Method and apparatus for determining preexisting stresses based on indentation or other mechanical probing of a material

ABSTRACT

Provided are methods and apparatus for determining from indentation testing the preexisting stress and/or effective strain in a section of a material. The invention also provides methods and apparatus for determining the variation of the stress with depth in the material (e.g. the gradient). According to the invention, first data are provided from an indentation test of the stressed (or strained) section. The stress (or effective strain) can then be determined from the first data and from second data characteristic of the material, such as a stress-strain curve. Second data can also be obtained from an additional indentation test of a section having a known stress. The methods provided herein are suitable for programming on a general purpose computer or calculator.

FIELD OF THE INVENTION

The present invention relates generally to the testing of materials, andmore particularly to indentation testing with the purpose of determiningthe stress in a stressed section of a material.

BACKGROUND

Known in the art are techniques whereby mechanical tests that generateload-penetration (P-h) curves are performed on a material to determineparameters that characterize the material, such as Young's modulus (E),yield strength (σ_(y)), stress at 29% plastic strain (σ_(u)), andhardness. See for example, WO 97/39333, published Oct. 23, 1997 andherein incorporated by reference, summarizing many known techniques inthe Background section, and disclosing refinements thereof. However,though the above properties are important, and the known techniques arehelpful in quickly characterizing materials, also of considerableimportance is an understanding and evaluation of the preexistingstresses present in a material or structure.

Many common industrial processes create, typically, though not always,as an undesirable side effect, stresses in materials. For example,stresses are induced when cooling a material from a processingtemperature; when depositing a coating or a thin film of a material on asubstrate by any of a number of techniques, such as chemical vapordeposition (CVD), molecular beam epitaxy (MBE), thermal spray,sputtering, or evaporation; when shot peening, laser shot peening,bending or loading a material; when a material undergoes a phasetransformation; and when welding materials together. The above representjust a few of the common industrial processes that can result instresses being induced in the material in question.

Whether stresses are intentionally created or are, as noted above, anundesirable side effect, it almost always desired to quantitativelycharacterize them stresses. As is appreciated by one of ordinary skillin the art, an understanding of these stresses can be important indetermining likely failure modes, in assessing the cause of existingfailures, in lifetime analyses, in quality control, and in a myriad ofother applications. Based on a knowledge of preexisting stresses,process variables can be adjusted to optimize a particular process, anddesigns changed to avoid potentially damaging residual stresses.Understanding and evaluating such stresses can be important in assessingthe integrity of structures having dimensions that range from ananometer scale to a micrometer and to a macrometer scale.

For example, in the production of integrated circuits, electricallyconductive vias are required to provide electrical communication betweenotherwise isolated circuit layers. To produce a via, the electricallyconductive material is deposited in an etched hole. Failure of such viasis known to cause performance reductions or outright failure ofintegrated circuits, and ready evaluation of the stress in vias would bean important tool in screening defective chips or in optimizingdeposition parameters to avoid the potentially damaging stresses. Yet nosimple and economical technique is available that is reliable, fast andsuitable for the physical scale of the vias (typically microns) and foruse in a high volume production environment. Known techniques, such ashole drilling, are destructive, and can be particularly inappropriate.

Approaching the other end of the size scale, knowledge of the stressescan be of immeasurable benefit in determining, for example, theintegrity of the numerous welds in the labyrinth of piping required in anuclear power plant, and, on the macro scale, in assessing the integrityof submarine hulls.

Despite the importance of understanding and evaluating stresses, to theinventors knowledge there exists few methods or apparatus forquantitatively determining the preexisting stresses in a material basedon an indentation test. Other known methods, which include holedrilling, layer removal, strain, displacement or curvature measurements,X-ray diffraction or neutron diffraction, are typically tedious and/orexpensive, and often not suitable for economically and quickly testingproducts, especially in the large volume production environment notedabove. Many of the methods, such as hole drilling and layer removal, canrender the test sample unsuitable for further use, and cannot beperformed on any large number samples. Unfortunately, known techniquescan be more likely to be performed after, rather than before, acatastrophic failure, such as to determine why a tank car derailed, orwhy a turbine blade failed and destroyed an aircraft engine, and toasses liability of manufacturer.

Recent efforts, such as those of Tsui et al. (1996) and Bolshakov et al.(1996) are empirical observational studies that report no general orspecific formulation for determining stress. The only major outcome ofthese studies is the realization that the overall hardness and theelastic modulus of an elastoplastic material may not be affected by anypre-existing elastic residual stress field. See Tsui, T. Y., Oliver, W.C. and Pharr, G. M (1996) "Influences of Stress on the Measurement ofMechanical Properties Using Nanoindentation: Part I. Experimentalstudies in an aluminum alloy", J. Mater. Res., vol. 11, pp. 752-759.,herein incorporated by reference, and Bolshakov, A., Oliver, W. C. andPharr, G. M (1996) Influences of Stress on the Measurement of MechanicalProperties Using Nanoindentation: Part II Finite Element Simulations,"J. Mater. Res., vol. 11, pp. 752-759.

Accordingly, as improved techniques and apparatus for determiningstresses would be a welcome advance in the art, it is an object of thepresent invention to address one or more of the foregoing disadvantagesand drawbacks of the prior art.

It is a further object of the invention to provide methods and apparatusfor allowing a simple mechanical test for determining the stresses inmaterials.

Other objects will be in part be apparent and in part appearhereinafter.

SUMMARY OF THE INVENTION

The invention achieves the foregoing objects by providing method andapparatus for determining preexisting stresses in a stressed section ofmaterial based of data obtained from indentation testing of the stressedsection. As used herein, the term preexisting stresses refers toresidual, internal, or applied stresses. Indentation testing refersgenerally to probing wherein a force is applied to a section of amaterial to obtain data such as the load on the material, the projectedcontact area between the force applying probe and the material, and thedepth of penetration. Data can also include a load-depth (P-h) unloadingand loading curves, and the slope of the curves, particularly the slopeof an unloading curve. An unloading slope can be obtained by a slightunloading of the indenter during loading. Examples of indenters includea servo hydraulic element; a screw-drive indenter; a hardness tester; amicroindenter; a nanoindenter; and an atomic force microscope.

According to the invention, the existence of hardness invariance and theeffect of stresses on an indentation are appreciated and understood suchthat methods and apparatus are disclosed for determining stresses in astressed section of a material. The stresses are determined from asimple indentation test performed on the stressed section of materialand from a knowledge of other properties that are characteristic of thematerial.

As used herein, the phrase "properties characteristic of the material"refers to properties that characterize the material, such as Young'smodulus, yield strength, stress at 29% plastic strain, strain hardeningexponent, and hardness, or a stress strain curve for the material, whichas understood by one of ordinary skill, is determined from a knowledgeof the above parameters. The term also includes measurements done, suchas indentation tests, on a section having a known stress of the materialor a section having a known stress of a material substantially similarto the material. As understood by one of ordinary skill, and disclosedfor example in WO 97/39333 incorporated by reference above, at least oneof the foregoing parameters can be determined from such tests.

In practicing the invention, first data are typically obtained from anindentation test of a prestressed section of material. The first datacan include a depth of penetration hs, a load on the indenter, Ps, andan area of indentation, As. Second data characteristic of the materialare then obtained, from any one of a variety of sources, such asmechanical tests including uniaxial compression test, such as anindentation tests, graphs or tables. Second data can also be obtainedfrom a manipulation of the first data, such as by dividing a measuredload on the indenter Ps by the area of indentation, As, to obtain thehardness, which, as disclosed herein, is invariant. The stress is thendetermined in accordance with the present invention from the first andsecond data, as is disclosed below.

In one aspect of the invention, an analogy is drawn between the problemof determining strain in an plastically graded material and the problemof determining stress such that methods and apparatus are disclosed fordetermining the stress in a material, including spatial variation of thestress.

In one aspect of the invention, methods and apparatus are disclosed fordetermining the stresses in a material wherein the stresses can beconsidered constant, either because the stresses are in factapproximately constant with penetration, or because the purpose to whichthe stress data are to be used allow such an approximation. In anotheraspect of the invention, methods and apparatus are disclosed fordetermining the variation of the stress with depth, or position, intothe stressed section of material.

In a further aspect of the invention methods and apparatus are disclosedfor determining the residual yield strength and in turn the equivalentplastic strain from an indentation test of section that has beenplastically yielded.

According to one feature of the invention, it is appreciated that theYoung's modulus of the material is substantially invariant to stressesand strains induced in the material, allowing additional data to bedetermined from a mechanical test, such as an indentation test, of thematerial, such that fewer properties characteristic of the material needbe known from other sources or tests to determine the stresses andequivalent plastic strains.

Each of the method and apparatus broadly described above are nextdescribed in turn and in detail

According to one feature of the invention, a method for determining thepreexisting stress in a stressed section of a material includesobtaining first data from the indentation of the stressed section of thematerial with an indenter, the first data including at least one of aload on the indenter P_(s) and an area of indentation A_(s) ; obtainingsecond data characteristic of the material; and determining from thefirst and second data the stress in the stressed section of material.

According to another feature of the invention, a method for determiningthe preexisting stress in a stressed section of a material includesobtaining first data from the indentation of the section of materialwith an indenter, the first data including a load P_(s) and an areaA_(s) at the load P_(s) ; obtaining second data from the indentationwith a second indenter of one of a section having a known stress of thematerial and a section having a known stress of a first materialsubstantially similar to the material, the second data including atleast an area Ao at a load Po substantially equal to P_(s) ; anddetermining the stress in the stressed section of the material fromP_(s), P_(o), A_(o) and A_(s).

According to a further feature of the invention, a method of determiningthe stress in a stressed section of a material includes obtaining firstdata from the indentation of the stressed section with an indenter, thefirst data including at least a load Ps and the correspondingindentation area As and penetration hs; obtaining second datacharacteristic of the material, the second data including Young'smodulus (E), the yield strength (σy), and one of the stress atapproximately 29% plastic strain (σu) and the strain hardening exponent(h); determining the hardness Pave from one of Ps/As and the seconddata; determining (P_(o)) from P_(o) =(h_(s))² σy (1+σu/σy)C*{1+ln[(tanα)E/3σy]} (where C* and tan a depend on the type of the indenter);determining A_(o) from A_(o) =P_(o) /Pave; determining the ratio R=A_(s)/A_(o) ; determining the stress in satisfaction one of the followingformulae

    if R<1; R={1+stress/Pave}.sup.-1

    if R>1; R={1-(geomf)stress/Pave}.sup.-1

where geomf=sin(α), and α is related to the angle of indentation of theindenter.

According to a further feature of the invention, a method of determiningthe stress in a stressed section of a material includes obtaining firstdata from the indentation of the stressed section with an indenter, thefirst data including at least a load Ps and the correspondingindentation area As and penetration hs; obtaining second datacharacteristic of the material, the second data including Young'smodulus (E), the yield strength (σ_(y)), and one of the stress atapproximately 29% plastic strain (σ_(u)) and the strain hardeningexponent(h); determining the hardness Pave from one of Ps/As and thesecond data; determining (h_(o))² =P_(s) {σy(1+σu/σy)C*[1+ln((tanα)E/3σy)]}⁻¹ (where C* and tan a depend on the type of the indenter);determining the ratio R=(h_(s))² /(h_(o))² ; determining the stress insatisfaction of the following formulae;

    if R<1; R={1+(geomf)stress/Pave}.sup.-1

    if R>1; R={1-stress/Pave}.sup.-1

where

geomf=sin.sub.α, and

α is related to the angle of indentation of the indenter.

According to yet another feature of the invention, a method ofdetermining the stress in a stressed section of a material includesobtaining a loading curve of load P verses penetration h from theindentation with an indenter of the stressed section; fitting theloading to curve to a polynomial expression of the form B,h² +B₂ h³ todetermine first and second constants B1 and B2; based on the knownproperties of the indenter determining at least one additional constantB₃, from B₁ ; obtaining a value for the yield strength (σy)characteristic of the material; determining the stress G in the sectionof material as a function of the at least one additional constant B₃ andσy.

According to an additional feature of the invention, a method ofdetermining stress at the surface of a stressed section of a materialand of determining the variation of stress with penetration includes:obtaining a loading curve of load P versus position h from theindentation with an indenter of the section of material; fitting theloading curve to a polynomial expression B₁ h² +B₂ h³ to obtainconstants B₁ and B₂ ; determining third and fourth constants B₃ and B₄in satisfaction of the formulas; B₃₌ B₁ /[11.88(tan g)² ] and B₄ =B₂/[8√3(tan γ)³ ], where g is a known angle of the indenter; obtaining avalue for the yield strength (σy) of the material; determining at leastone of the magnitude of the stress at the surface of the section ofmaterial, G, and the magnitude of the rate of change of the stress withpenetration, Γ, in satisfaction of the formulas B₃ =-GΓ and B₄ =(σy)²-G². In another feature of the invention, a method of determining theeffective plastic strain in a plastically-strained section of a materialincludes obtaining first data from the indentation of the section withan indenter, the first data including a penetration of (h) and a load(P) on the indenter; obtaining second data characteristic of thematerial; determining the residual yield strength as a function of thesecond data and the first data; and determining the effective plasticstrain from the residual yield strength.

The invention also includes apparatus for determining preexistingstresses and/or or preexisting plastic strains in a stressed section ofa material.

According to one feature, the invention provides apparatus fordetermining the preexisting stress in a stressed section of material.The apparatus includes a data processor, where the data processorincludes program element for determining the stress in a stressedsection of a material from first data and from second datacharacteristic of the material, the first data obtained from anindentation test on the stressed section and including at least one of aload on the indenter P_(s) and an area of indentation A_(s). The termdata processor, as used herein, is intended to include a computer, acalculator, and a suitable general purpose or dedicated integratedcircuit. In yet another feature, the invention provides an apparatus fordetermining the preexisting stress in a stressed section of a material,the apparatus including a data processor, the data processor includingprogram element for determining the stress in a stressed section of amaterial from first data and from second data characteristic of thematerial, the first data obtained from an indentation test on thestressed section and including at least an loading P-h curve and thesecond data including the yield strength of the material, the dataprocessor further including an element for fitting the loading to curveto a polynomial expression of the form B₁ h² +B₂ h³ to determine atleast first and second constants B₁ and B₂ ; an element for determiningat least one additional constant B₃ from B₁ ; and an element fordetermining the stress G in the section of material as a function of theat least one additional constant B₃ and the yield strength σy .

In an another feature, the invention provides apparatus for determiningthe plastic strain in a section of material that has plasticallyyielded. The apparatus includes a data processor, the data processorincluding: an element for determining from first data obtained byindenting the section with an indenter and from second datacharacteristic of the material the stress in the stressed section, thefirst data including a penetration (h) and a load (P) on the indenter,the element including element for determining the residual yieldstrength as a function of the second data and the first data; andelement for determining the plastic strain in the section of thematerial from the residual yield strength and the second data.

The invention also includes a data processor program product fordetermining the preexisting stress in a stressed section of material,the data processor program product including a medium readable by a dataprocessor; and data processor program logic recorded in the dataprocessor readable medium and executable by a data processor to defineelement for determining the stress in a stressed section of a materialfrom first data and from second data characteristic of the material, thefirst data obtained from an indentation test on the stressed section andincluding at least one of a load on the indenter P_(s) and an area ofindentation A_(s).

The invention also includes a data processor program product fordetermining the preexisting stress in a stressed section of material,the data processor program product including a medium readable by a dataprocessor; and data processor program logic recorded in the dataprocessor readable medium and executable by a data processor to definean element for determining the stress in a stressed section of amaterial from first data and from second data characteristic of thematerial, the first data obtained from an indentation test on thestressed section and including at least an loading P-h curve and thesecond data including the yield strength of the material, the elementfurther including an element for fitting the loading to curve to apolynomial expression of the form B₁ h² +B₂ h³ to element fordetermining at least first and second constants B₁ and B₂ ; an elementfor determining at least one additional constant B₃ from B₁ ; and anelement for determining the stress G in the section of material as afunction of the at least one additional constant B₃ and the yieldstrength σy. The data processor program logic can also include anelement for determining the change of stress with depth into thematerial.

Additionally, the invention includes a data processor program productfor determining the plastic strain in a section of material that hasplastically yielded, including: a medium readable by a data processor;and data processor program logic recorded in the data processor readablemedium and executable by a data processor to define: an element fordetermining from first data obtained by indenting the section with anindenter and from second data characteristic of the material the plasticstrain in the section, the first data including a penetration (h) and aload (P) on the indenter, the element including an element fordetermining the residual yield strength as a function of the second dataand the first data; and an element for determining the plastic strain inthe section of the material from the residual yield strength and thesecond data.

Suitable apparatus for performing the indentation tests described hereinare known in the art, such as the apparatus disclosed in WO 97/39333,published Oct. 23, 1997, and herein incorporated by reference.

The methods disclosed herein are amenable to programming on and solutionby a suitable data processor, including a computer, such as an IBM typepersonal computer, a calculator, or a dedicated integrated circuit. Asunderstood by one of ordinary skill in the art, in light of thedisclosure herein, such a data processor can communicate directly withindentation apparatus for controlling the indentation of the stressedand/or unstressed sections of a material, and for processing data fordetermining the stress in the stressed section of material.Alternatively, the data processor can obtain data from input files thatinclude indentation data previously obtained. The apparatus of theinvention can include a data processor, such as a computer programmed todetermine stresses, as well as computer data products, such as memoryproducts, storing program elements for determining stresses in astressed section of material.

It is considered that the invention advantageously represents asignificant improvement in the field of stress strain evaluation, andcan have a considerable practical application in numerous fields ofendeavor in addition to those enumerated in the Background.

To the inventors' knowledge the invention provides the first method andapparatus for qualitatively determining the preexisting stress in astressed section of material, including the magnitude and the sign ofthe stress, as well as for qualitatively determining the magnitude andsign of preexisting plastic strains and the gradients in preexistingstresses, using indentation. The potential application for the presentinvention are virtually too numerous to list. Stresses can be introducedin a section of a material by industrial processes that include, thoughtare not limited to, machining, grinding, lapping, milling, turning,rolling, polishing, heat treatment, such as rapid quenching and casehardening, chemical etching, laser etching, laser ablation, laser shockpeening, shot peening ion implantation, extruding--the list is virtuallyendless. The induced stress can be an undesired side effect of theindustrial process, or alternatively, purposely introduced. Regardless,it is desirable and of considerable advantage to be able toquantitatively determine such stresses, as well as the gradients instresses and the plastic strain, according the methods and apparatus ofthe invention.

For example, it is considered that there are applications of theinvention in a wide variety of fields, for example, indenting todetermining the stresses and/or plastic strains in a film adhering to abiomechanical implant, thin metallic films deposited in formingmicroelectronics circuits and conductive vias in integrated circuits,welded joints, annealed materials, aircraft engine components, s such asturbine blades, automobile panels, airframes, submarine hulls--again,the list can almost be endless, and in each circumstance it isanticipated that engineers and designers will welcome and benefit fromthe ability to quickly and simply determine stresses and/or strainsaccording to the techniques and apparatus of the present invention.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 illustrates indentation apparatus suitable for indenting astressed section of a material in accordance with practicing theinvention;

FIG. 2 illustrates typical loading and unloading P-h curves obtainedfrom an indentation test using the indentation apparatus of FIG. 1;

FIG. 3 illustrates a typical uniaxial stress strain curve for a materialindented by the indentation apparatus of FIG. 1;

FIG. 4 schematically illustrates indentation of the stressed section ofthe material with the indenter of FIG. 1 and the nature of the stressesin the stressed section of the material;

FIG. 5 is a stress-strain curve graphically illustrating the effect ofplastic strains in the stressed section of material indented by theindentation apparatus of FIG. 1 on the yield behavior of the stressedsection;

FIG. 6 schematically illustrates the effect of tensile stress in thestressed section of material on the indentation of the stressed sectionof material by the indentation apparatus of FIG. 1;

FIG. 7A schematically illustrates the effect of compressive stress inthe stressed section of material on the indentation of the stressedsection of material by the indentation apparatus of FIG. 1;

FIG. 7B illustrates the angle of indentation of an indenter having afinite tip radius.

FIG. 8A graphically illustrates the variation of the area of indentationas a function of the stress in the stressed section of material, and acomparison between finite element analysis predictions and the methodsof the present invention;

FIG. 8B graphically illustrates the variation of the area of indentationas a function of the stress in the stressed section of material, and acomparison between empirical data and the area of indentation asdetermined in accordance with the methods of the present invention;

FIG. 9 graphically illustrates indentation P-h curves for a section ofthe material without stress and for the stressed section undercompressive and tensile stress, the initial unloading slope for eachcurve, and the determination of stress in the stressed section accordingto an equal penetration strategy;

FIG. 10 is similar to FIG. 9 and illustrates determination of thestressed in the stressed section according to an equal load strategy;

FIGS. 11A and 11B illustrate the analogy between the determination ofstress as a function of penetration in the stressed section and thedetermination of residual yield strength as a function of depth in anelastoplastically graded material;

FIG. 12 illustrates indention P-h curves characteristic of a section ofthe material without stress and for the stressed section under tensileand compressive stresses that vary with depth into the material;

FIG. 13 illustrates schematically apparatus of the invention includingan indentation apparatus mounted in a load applying frame and a computersystem for collecting and processing data for determining stress;

FIG. 14 is a block diagram of an exemplary computer system forprocessing indentation data and data characteristic of the materialindented so as to determine stress in accordance with the invention;

FIG. 15 is a block diagram of the memory system shown in FIG. 15;

FIG. 16 is a high-level flow chart illustrating the determination thestresses in the stressed section of material by a computer systemexecuting program instructions according to the invention, such as thecomputer system of FIG. 14.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENT

The present invention is directed to determining the preexistingstresses in a stressed section of a material from an indentation test ofthe stressed section. The stresses may be residual stresses created byany number of processes, such as cooling a material from a processingtemperature; depositing a coating or a thin film of a material on asubstrate by any of a number of techniques, such as chemical vapordeposition (CVD), molecular beam epitaxy (MBE), thermal spray,sputtering, or evaporation; shot peening, laser shot peening, bending orloading a material; causing a material to undergo a phasetransformation; and welding materials together.

The invention provides methods and apparatus that readily scale todetermine stresses in variety of material and environments, ranging fromnanometer to micro and macrometer measurements. For example,applications can include determining the stresses on etched lines ofthin films deposited on an electronic substrate, to underwater tests onthe hull of a submarine.

Many types of indentation apparatus are known in the art. FIG. 1illustrates one known indentation apparatus, indicated generally by thereference numeral 20. The indentation apparatus 20 includes a loadapplying frame 22. A load cell 24 couples an upper mount 26 to anindenter mount 28 which mounts an indenter 30. An adjusting mountassembly 32 couples a base support 34 to a mounting fixture 36 having anupper surface 38 for mounting the material 40 such that the stressedsection 42 of the material 40 is indented by the indenter 30. Theindentation apparatus 20 also includes a displacement, or penetration,sensor 46 and a mirror 48 with which the sensor 46 cooperates todetermine the penetration of the indenter 30 into the material 40. Theindentation apparatus can optionally include an area measurement device50, schematically illustrated in FIG. 1, for determining the projectedarea of indentation. As is known by those ordinary skilled in the art,the projected area of indentation may be measured during or afterindentation, and the area measurement device can include, though is notlimited to, an optical device, a refractive device that useselectromagnetic radiation, or a surface profilometer. The areameasurement device need not be mounted as with the indenter mount 28, ascan be located adjacent the indentation apparatus 20, as indicated bythe dotted box 50'.

The indentation apparatus 20 can therefore generally provide three typesof first data pertaining to the stressed section 42: The load on theindenter 30; the projected area of indentation as measured by the areameasurement device 50 or by other techniques described below; and thepenetration of the indenter 30 into the sample, as measured by thesensor 46 operating cooperatively with the mirror 48. The letter "P" isused herein to signify the load; the "h" is used herein to signify thepenetration of the indenter into the material 40; and the letter "A" isused herein to signify the projected area of indentation, which canalternatively be determined by a measurement of the slope of P versus Hupon initial unloading, as is further described below and as isdescribed in WO 97/39333 published Oct. 23, 1997 and herein incorporatedreference. The indentation apparatus can provide complete unloading andloading P-h curves, just one of the curves, or just a set of data points(e.g.. an hs and a Ps, where subscript s refers to the stressed section42). It is not required that an indentation provide all three types ofdata to determine the stress in the stressed section 42.

FIG. 2 graphically illustrates the load and displacement data typicallyobtained from an indentation apparatus such as indentation apparatus 20.The load P is plotted versus the penetration h. At point 54 the maximumload on the indenter 30 P_(s)(max) corresponds to a maximum penetrationh_(s)(max) indicated by reference number 56. Both a loading curve 52 andan unloading curve 53 can be obtained. At each point along each of thesecurves there is a load, such as a load P_(s) that corresponds to apenetration, such as penetration h_(s) and an area A_(s). The point 62indicates Ps and hs.

The slope of the unloading curve 53 upon initial unloading,dP/dh|P(smax), is indicated by reference numeral 58. Reference numeral60 refers to the residual penetration hr upon unloading, indicating thatthe material 40 remains penetrated by the indenter 30 to a residualdepth h_(r).

The data obtained from an indentation test of the material 42 need notinclude both a loading curve 52 and an unloading curve 53 or acomputation of the slope 58. As will be understood by one of ordinaryskill in the art, based on the subsequent disclosure herein, determiningthe stress in the stressed section 42 of the material 40 does notnecessarily require a complete P-h curve. In some instances, data thatincludes two of P_(s), h_(s) and A_(s) can suffice. Considerable detailis provided below on determining the stresses based on data obtainedfrom an indentation test performed by the indentation apparatus 20.

It is known in the art to perform indentation test on an unstressedsection of a material to obtain for example, Young's modulus (E), theyield strength (σy), and the stress at approximately 29% plastic strain(σu), or the strain hardening exponent, h. Such quantities define astress-strain curve for a material. With reference to FIG. 3, referencenumeral 64 indicates a typical stress strain curve for a material.Young's modulus (E) refers to the linear portion 66 of the curve 64. Theyield strength of the material s_(y) indicates the point 68 where thematerial no longer behaves linearly, that is, the stress is no longer alinear function of the strain. The stress at approximately 29% plasticstrain indicated by reference numeral 70 is equivalent to characterizingthe exponential behavior of the stress-strain according to the strainhardening exponent h. Such exponential behavior is demonstrated by theportion of the stress strain curve 64 indicated generally by thereference numeral 72.

The following discussion of FIGS. 4 and 5 is intended to provide ageneral framework for the more detailed disclosure below of practicingthe invention to determine the stresses and/or strain in the stressedsection 42.

The invention is directed to determining stresses in the section 42,including the variation of stress with penetration h, and to determiningresidual yield strength σ_(y) ^(R) and residual plastic strainε_(e),0^(R) when the stressed section 42 has plastically yielded.

With reference to FIG. 4, the indenter 30 indents the surface 76 of thestressed section 42 with a load P, indicated by reference numeral 74.The stressed section 42 is considered to be subject to an equal-biaxialstate of residual stress (tensile or compressive) whose magnitude at thesurface is σ_(x),0^(R) =σ_(y),0^(R). The equi-biaxial elastic residualstrain at the surface 76 can be defined as ##EQU1## where u is thePoisson ratio of the material 40. The residual field at the surface 76can be elastic or plastic. If only elastic residual stresses are presentat the surface 76, then |σ_(x),0^(R) |=|σ_(y),0^(R) |≦σy where σy is theinitial (reference) yield strength of the substrate at the surface.

If, on the other hand, the surface 76 with the residual field isplastically yielded, a new yield strength σ_(y) ^(R), is ascribed to thesurface 76. The new yield strength σ_(y) ^(R) accounts for thepre-existing equi-biaxial plastic strain at the surface, ε₀ ^(pl). Notethat for the equi-biaxial plastic deformation at the surface 76,ε_(x),0^(pl) =ε_(y),0^(pl), with the von Mises effective yield strain,ε_(e),0^(pl) defined as ##EQU2##

It is easily shown that ε_(e),0^(pl) =2|ε_(x),0^(pl) |=2|ε_(y),0^(pl) |.Conservation of volume during plastic deformation requires the followingcondition to be satisfied: ε_(e),0^(pl) =2ε_(x),0^(pl) =2ε_(y),0^(pl).Similarly, note that the equi-biaxial surface residual stress,σ_(x),0^(R) =σ_(y),0^(R) =σ_(e),0^(R), where σ_(e),0^(R) is the vonMises effective stress as defined as ##EQU3##

As a consequence of Equations (2) and (3), the yield strength of thepreviously plastically deformed surface of the substrate containing aresidual plastic strain can be linked to the plastic strain by theequation:

    σ.sub.e,0.sup.R =σ.sub.y.sup.R =F(ε.sub.e,0.sup.p.spsp.1) F(ε.sub.e,0.sup.p.spsp.1)=A(ε.sub.e,0.sup.p.spsp.1).sup.n( 4)

where A is an experimentally determined material constant, and h is thestrain hardening exponent. For the biaxial stress state representativeof the surface residual stress or strain, the function F in Equation (4)mirrors a plot as shown in FIG. 5, of the uniaxial stress versusuniaxial plastic strain, indicated by reference numeral 80, for a simplecompression test. See, for example, Nadai, A. I. (1963) Theory of Flowand Fracture of Solids, vol. II, McGraw-Hill, N. Y., herein incorporatedby reference. Typically, the sign of the residual stress cannot beidentified from the definition of the effective stress or strain, asgiven in Equations (2) and (3). The sign of the residual plastic strain,however, can be determined from the sign of the elastic residual strain,as described below and/or from a knowledge of prior deformation historyof the indented surface 76 (e.g., shot peening or bending on thecompression side) or from indentation tests, as disclosed below.

An Approach to the Determination of Preexisting Stresses.

Denote by P₀ and A₀ the load and area of indentation corresponding to aindentation of a section without stress of the material 40. For the samemaximum indentation depth, let the corresponding load and contact areaof an indentation of the stressed section 42 be denoted by P and A. Fromthe invariance of hardness or equivalently the average contact pressurewith respect to the contact area A, penetration depth h, applied to loadP or the elastic residual stresses, it is seen that ##EQU4## providedthat indentation is performed to the same depth. If the material 40 ispurely elastic, then the contact pressure remains invariant irrespectiveof the magnitude of the residual stress, or the size and shape of theindenter, or the magnitude of the applied load. If the residual stressfield (either spatially constant or varying) is elastic and if thesubstrate is elastic throughout indentation, then there is no effect ofthe residual stress on the indentation response (i.e. P-h curves,contact areas, average pressures, and stress and strain fields areunaffected by the residual stress).

Equation (6) relates the indentation loads directly to the contactareas, and has been substantiated by experimental observations. seeTsui, T. Y., Oliver, W. C. and Pharr, G. M (1996) "Influences of stresson the measurement of mechanical properties using nanoindentation: PartI. Experimental studies in an aluminum alloy", J. Mater. Res., vol. 11,pp. 752-759., herein incorporated by reference.

According to the invention, a relationship between the contact areasA_(o) and A is formulated such that the stress in the section 42 can bedetermined.

With reference to FIG. 6, consider indentation of the section 42 whenthe section 42 is subject to a tensile stress.

An equi-biaxial tensile residual stress at the indented surface,σ_(x),0^(R) =σ_(y),0^(R), can be considered equivalent to a tensilehydrostatic stress, σ_(x),0^(R) =σ_(y),0^(R) =σ_(z),0^(R) =σ_(H) plus auniaxial compressive stress component -σ_(z),0^(R), as shown in FIG. 6.The hydrostatic stress, σ_(H), has no effect on the indentation-induceddeformation, and hence the average indentation pressure, _(Pave), isunaffected by σ_(H). Therefore, for a given applied load, P, the neteffect is a local enhancement of indentation force by an amount σ_(H) A.That is, given the presence of an equi-biaxial tensile residual stress,the indentation force is apparently enhanced from P to P+σ_(H) A.Consequently, an indented section 42 with a tensile equi-biaxialresidual stress will develop a larger contact area compared to aninitially stress-free surface when indented to the same load P. We thussee that

    P.sub.apparent =P.sub.0 +σ.sub.H A.                  (7)

Therefore,

    .sub.Pave A.sub.0 =(.sub.Pave -σ.sub.H)A.            (8)

For a fixed applied load P, this equation can be rewritten as ##EQU5##

If, instead, a fixed indenter penetration depth, h, is considered, notethat

    P=P.sub.0 -σ.sub.H A=P.sub.0 -σ.sub.x,0.sup.R A=P.sub.0 -σ.sub.y,0.sup.R A.                                 (9b)

Physically, this implies that when the section 42 is subjected to atensile biaxial residual field, the load P necessary to indent to adepth of penetration h is smaller than that required for a stress-freesubstrate indented to the same depth h. This drop in the indenter loadnecessary to induce a fixed depth of penetration h in the presence of atensile elastic residual stress field results in a correspondingreduction in the contact area. Invoking now the invariance of Pave, itcan be readily shown that the reduced contact area A under fixed h is:##EQU6##

With reference to FIG. 7, consider now that the section 42 is subject toa compressive stress, -σ_(x),0^(R) =-σ_(y),0^(R). In this case, thebiaxial residual stress would be equivalent to a compressive hydrostaticstress, -σ_(x),0^(R) =-σ_(y),0^(R) =-σ_(z),0^(R) =-σ_(H) plus a uniaxialtensile stress component σ_(z),0^(R). Because this fictitious uniaxialtensile stress of constant magnitude could cause loss of contact at thecontact perimeter (since it acts counter to the direction of theindentation load P), Equations (8) and (9) are preferably not directlyextended here with a simple change of sign for σ_(H). To reduce error inthe subsequent determination of stress, the following approach isconsidered preferable.

The component of the residual compressive stress which facilitatescontact between the indenter 30 and the surface 76 acts normal to theinclined faces 85 of the indenter 30, and is indicated by referencenumeral 86. Its magnitude, as shown in FIG. 7 is σ_(x),0^(R) sinα=σ_(y),0^(R) sin α, where α=π/-γ with 2γ being the included angle ofthe indenter tip, where γ is referred to herein as the angle ofindentation of the indenter 30. (Recall that α=22° for the Vickerstetragonal pyramid indenter, 24.7° for the Berkovich trigonal pyramidindenter, and 19.7° for the equivalent circular indenter.) FIG. 7 alsoillustrates determining the angle γ when the indenter is characterizedby a tip radius r. As illustrated in FIG. 7, a part of the applied load,equal in magnitude to (σ_(x),0^(R) sin α)×A, where A refers to thecontact area 85, is expended in creating a hydrostatic stress which doesnot contribute to the overall hardness. Thus, for a given applied loadP, the effective indentation load is smaller, and consequently, theindentation stiffness is apparently raised, making it harder topenetrate the material in the presence of a compressive residual stressfield. The invariance of the contact pressure for the cases with andwithout surface residual stresses, is again invoked:

    .sub.Pave A.sub.0 =(.sub.Pave +σ.sub.x,0.sup.R sin α)A=(.sub.Pave +σ.sub.y,0.sup.R sin α)A. (10a)

This equation readily gives the area ratio ##EQU7##

A compressive equibiaxial residual stress at the surface 76 aidsindentation such that the contact area A (and equivalently, the depth ofpenetration, h) for a given load P, is smaller than that for a surfacewithout any residual stress.

Repeating the reasoning given in connection with the derivation ofEquation (9c), for a fixed h, the contact area ratio in the presence ofan equi-biaxial elastic compressive residual stress field becomes##EQU8##

In other words, the compressive residual field enhances the indentationload to produce a fixed h (compared to that for a stress-freesubstrate). Consequently, the contact area increases.

The elastic response during indentation is fully independent of anypre-existing residual stresses at the surface. Therefore, the initialunloading portion of the P-h curve in instrumented indentation would beexpected to be unaffected by residual stresses.

Equations 9 and 10 have been verified by finite element analysis and bycomparison to published experimental data. Tables I and II show thecomparison of the finite element modeling with results calculated inaccordance with equations 9 and 10.

                  TABLE I                                                         ______________________________________                                        Comparison of finite element and analytic results. Sharp indentation          of an elastic-perfectly plastic surface with equi-biaxial plastic             prestrain.                                                                               ε.sub.e.sup.p = 2 |ε.sub.x.sup.p                     |  p.sub.av /σ.sub.y                                                                 (tan γ = 0.3)                          E/σ.sub.y                                                                          (ε.sub.x.sup.P = ε.sub.y.sup.p)                                           Theory    FEM                                          ______________________________________                                        200        ±0.045*  3         2.82                                         (>150)     ±0.095   3         3.09                                         100        ±0.0425  1.93      1.84                                         (<150)     ±0.09    1.93      1.80                                         ______________________________________                                         *+ for tensile,  for compressive residual strain                         

                  TABLE II                                                        ______________________________________                                        Comparison of finite element and analytic results. Sharp indentation          of an elastic-strain hardening plastic surface with equibiaxial plastic       prestrain. (linear, isotropic strain hardening)                                         ε.sub.e.sup.p = 2 |ε.sub.x.sup.p                     |  P.sub.av /σ.sub.y (tan γ = 0.3)             E/σ.sub.y                                                                        E.sub.T /E                                                                           (ε.sub.x.sup.p = ε.sub.y.sup.p)                                             Theory                                                                              FEM                                       ______________________________________                                        100      0.01   ±0.0425*   2.76  2.64                                      (<150)   0.01   ±0.09      2.46  --                                        200             ±0.045     2.84  2.85                                      (>150)          ±0.095     3.00  3.15                                      ______________________________________                                         *+ for tensile,  for compressive residual strain                         

FIG. 8A is a plot of the ratio ##EQU9## versus stress for a constantpenetration depth h according to equations 9 and 10 and according to afinite element analysis. The details of the finite element analysis aregiven in Giannakopoulus, A. E., Larsson, P.-L. and Verstergaard, R(1994) "Analysis of Vickers Indentation", International Journal ofSolids and Structures, vol. 31, pp. 2679-2708. herein incorporated byreference. Reference numeral 90 indicates a plot the ##EQU10## asdetermined by equations 9 and 10.

FIG. 8B is a comparison between results determined in accordance withequations 9 and 10 and experimental results. The experimentalnanoindentation results are for a rapidly solidified 8009 aluminum alloy(very fine-grained with room temperature yield strength, 353 MPa). Theseresults are taken from Tsui, T. Y., Oliver, W. C. and Pharr, G. M.(1996) "Influences of Stress on the Measurement of Mechanical PropertiesUsing Nanoindentation: Part I--Experimental Studies in an AluminumAlloy," J. Mater. Res., vol 11, pp. 752-759, herein incorporated byreference. The alloy was simultaneously subjected to different levels ofuniaxial as well as biaxial tensile or compressive elastic residualstresses while being indented by a Berkovich nanoindenter to a maximumload of 110 mN.

It is thus seen that the equations 9 and 10 are obtained from a soundphysical model of effect of stresses on an indentation test.

Thus, according to the invention, a relationship of the form

    R={1±(stress/Pave)geomf}.sup.-1                         (11)

is used for determining the stress in the stressed section 42 of thematerial 40 via an indentation test, such as with indentation apparatus20, of the stressed section 42. R is a ratio equivalent to the ratio ofthe indented area As of the stressed section 42 to an indented area Aocharacteristic of the material without stress (or, due to invariance ofhardness, R is equal to the ratio Ps/Po). Pave is the hardness that ischaracteristic of the material, (such as Po/Ao or Ps/As) and, asdiscussed above, geomf is an optional factor that is useful forimproving the accuracy of the stress determination and that is relatedto the shape of the indenter 30. Thus given Po, Ao, As and Ps, thestress can be determined.

As is discussed below, many strategies are possible for using equation11 for determining the preexisting stress in the stressed section 42.The ratio R need not be determined by actual indentation measurements ofstressed and unstressed sections of the material to determine the areasAs and Ao. A similar consideration applies to P_(ave) --it can, forexample be obtained from tabulated values, provided that the tabulatedvalues, if for example, for Rockwell or Brinell hardness number, areproperly converted to the Vickers hardness, or average pressure. In thefollowing discussion, and in the subsequent discussion of thedetermination of the variation of stress with penetration and of theplastic strain when the section 42 is plastically strained, reference ismade to the area of indentation As of the stressed section 42 of thematerial 40. The area As can be determined by the area measurementdevice 50 mounted with or adjacent the indentation apparatus 20. Asunderstood by one of ordinary skill in the art, there are many suitablearea measurement devices, such as, but not limited to, an opticaldevice, a refractive device (possibly optical, but more generally usingelectromagnetic radiation), and profilometer. However, as disclosedherein and in WO 97/39333 (PCT/US97/06425), an area of indentation canalso be determined by measuring the slope dP/dh, upon initial unloadingof the indenter 30. Such a determination of area is deemed within thescope of the invention, and can, in some instances, obviate the need forthe area measurement device 50.

As is understood by one of ordinary skill in the art, in light of thedisclosure herein, given a knowledge of data characteristic of thematerial, such as a stress strain curve, or of the Young's modulus, theyield strength, and the stress at approximately 29% plastic strain (or,equivalently, the strain hardening exponent) one or more of Ps, Po, Asand Ao can be determined from the other of the parameters Ps, Po, As andAo. Several actual examples strategies for determining stress are givenbelow will serve to illustrate the above.

To facilitate ease of understanding, many of the strategies fordetermining stress can be generally classified as either an "equal-load"approach or an "equal penetration" approach.

FIG. 9 is a graphical illustration of P-h curves and is useful inunderstanding an "equal penetration" approach. Curve 98 represents a P-hcurve of a section substantially free of stress of the material 40. Thesubstantially stress-free section is indented to a maximum load, denotedas P_(o)(max), to a maximum penetration, denoted as h_(o)(max)Indentation to a penetration h equal to ho of the stressed section 42that is subject to a compressive stress results in the dotted curve 102,which is above the substantially stress-free curve 98. Conversely,indentation to a penetration h equal to ho of the stressed section 42when the section 42 is subject to a tensile stress yields the P-h curve104 that resides below the substantially stress-free curve 98. Thuscomparison of P-h curves can allow a determination of the sign of thestress in the stressed section, that is, can allow a determination ofwhether the stress is compressive or tensile. Therefore, as discussedabove in the derivation of equations 9 and 10, or equivalently, notingthat As/Ao=P/Po, the ratio R is greater than 1. Similarly, for thetensile stress curve 108 as compared to the substantially stress freecurve 98, the ratio R is less than 1. Accordingly, Equation 11 above andequations 9 and 10 can be stated as follows for an "equal penetration"approach: ##EQU11##

FIG. 10 is a graphical illustration of P-h curves and is useful inunderstanding an "equal load" approach. Curve 98, 102 and 108 are asdescribed above with FIG. 9, namely, section 42 substantially free ofstress, under compressive stress, and under tensile stress,respectively. Again, the relationship of the curves can be used todetermine the sign of the stress. However, now the ratio R is related tothe points 110, 112 and 100, rather than points 106, 108 and 110.According, equation 9 and 10 can be stated as follows for the "equalload" approach: ##EQU12##

As is understood by one of ordinary skill in the art, in light of thedisclosure herein, P-h curves shown in FIGS. 9 and 10 can be obtainedvia indentation or calculation from properties characteristic of thematerial and those properties of the indenter necessary to make thecalculation in question. Properties of the indenter that are can beimportant include the type of the indenter (whether blunt or sharp), andthe Poisson ratio and Young's modulus of the indenter. For a indenter,whether the indenter is a Vickers, Berkovich, conical or rockwell type,and the angle of indentation g (or a, which is related to g) which areknown given the type of the indenter)

Several examples of strategies for determining the stress in thestressed section 42 are now given.

Strategy I

(1) indent the stressed section 42 to obtain first data including a loadPs and the corresponding penetration hs;

(2) obtain second data characteristic of the material, such as fromtables. The second data includes Young's modulus (E), the yield strength(σy), and the stress at approximately 29% plastic strain (σu) or thestrain hardening exponent.

(3) determine the hardness Pave from the second data.

(4) determine (h_(o))² where:

    (h.sub.o).sup.2 =P.sub.s {σy(1+σu/σy)C*[1+ln((tan α)E/3σy)]}.sup.-1                             (14)

where, as is known by one of ordinary skill in the art, C* is a constantrelated to the type of indenter.

(5) determine the ratio

    R=(h.sub.s).sup.2 /(h.sub.o).sup.2 ;                       (15)

(6) determine the stress in satisfaction of the equal load equation 13.

The foregoing strategy uses an indentation test of the stressed section42 and second data characteristic of the material, which can be obtainedfrom tabulated sources.

Strategy II

Obtain the second data for use as in Strategy I from indenting, usingany suitable indenter and indentation apparatus or other mechanical testframe, to indent a substantially stress free section of the material 40or of a material substantially similar thereto, to obtain the seconddata characteristic of the material 40.

Strategy III

(1) indent the stressed section 42 obtain first data including a loadP_(s) and the area A_(s) corresponding to the load P_(s) ;

(2) indent a section substantially free of stress of the material or asection substantially free of stress of a material substantially similarto the material to obtain second data that including at least an area Aoat a load Po substantially equal to P_(s) ;

(3) determine Pave=Po/Ao

(4) determine the stress in the stressed section in satisfaction of theequal load equation 13.

The foregoing approach uses two indentations to obtain data directly foruse in equation 13, and does not require a obtaining h from theindentation.

Strategy IV

(1) indent the stressed section 42 to obtain first data including a loadP_(s) and the area A_(s) and the penetration hs corresponding to theload P_(s) ;

(2) indent a section substantially free of stress of the material or asection substantially free of stress of a material substantially similarto the material to obtain second data that including area Ao and a loadPo at a penetration ho substantially equal to h_(s) ;

(3) determine Pave from Po/Ao or from Ps/As

(4) determine the stress in the stressed section in satisfaction of theequal penetration equation 12.

Strategy V

(1) indent the stressed section 42 to obtain first data including atleast a load Ps and the corresponding indentation area As anddisplacement hs;

(2) obtain second data characteristic of the material, including Young'smodulus (E), the yield strength (s_(y)), and the stress at approximately29% plastic strain (s_(u)) or the strain hardening exponent(h);

(3) determine the hardness Pave from Ps/As or from calculations from thesecond data;

(4) determine (P_(o)) from

    P.sub.o =(h.sub.s).sup.2 sy(1+σu/σy)C*{1+ln[(tan a)E/3σy]}(16)

where C*, as understood by one of ordinary skill, is a constant thatdepends on the type of the indenter.

(5) determine A_(o) from ##EQU13##

(6) determine the ratio R=A_(s) /A_(o)

(7) determine the stress in satisfaction of the equal depth ofpenetration equation

Strategy VI

Proceed as in strategy V, but obtain the second data from indenting,using any suitable indenter and indentation apparatus or othermechanical test, a substantially stress free section of the material 40or of a material substantially similar thereto, to obtain the seconddata characteristic of the material 40.

Strategy VII

(1) indent the stressed section 42 of the material 40 to obtain firstdata including Ps and hs

(2) obtain second data characteristic of the material, such as fromtables. The second data includes Young's modulus (E), the yield strength(s_(y)), and the stress at approximately 29% plastic strain (s_(u)) orthe strain hardening exponent. Obtain such second data that alsoincludes the hardness Pave, or a hardness number from which Pave can becalculated, or determine Pave from the Young's modulus (E), the yieldstrength (s_(y)), and the stress at approximately 29% plastic strain(s_(u)) or the strain hardening exponent.

(3) use Pave to obtain As, from As=Pave/Ps

(4) proceed as in step (4) and steps subsequent thereto in Strategy I oras in step (4) and steps subsequent thereto in Strategy V.

Strategy VIII

(1) indent the stressed section 42 of the material 40 to obtain firstdata including As and hs

(2) obtain second data characteristic of the material, such as fromtables and graphs. The second data includes Young's modulus (E), theyield strength (s_(y)), and the stress at approximately 29% plasticstrain (s_(u)) or the strain hardening exponent. Obtain such second datathat also includes the hardness Pave, or a hardness from which Pave canbe calculated, or determine Pave from the Young's modulus (E), the yieldstrength (s_(y)), and the stress at approximately 29% plastic strain(s_(u)) or the strain hardening exponent.

(3) determine Ps from Pave As

(4) proceed as in step (4) and steps subsequent thereto in Strategy I oras in step (4) and steps subsequent thereto in Strategy V.

The foregoing Strategies VII and VII requires fewer data to be takenfrom the indentation of the stressed section 42 of the material 40. Notethat the area As need not be measured by the measuring device 50 ordetermined from an unloading slope in strategy VII. Similarconsideration apply, but to Ps rather than As, for strategy VIII.

Strategy IX

(1) indent the stressed section 42 to obtain first data including a P-hcurve for both loading and at least initial unloading to obtain Psmax,and dPs/dhs

(2) indent a section substantially free of stress of the material 40 ora section substantially free of stress of a material substantiallysimilar to the material 40 to obtain a P-h curve including Pomax,including initial unloading to obtain dPo/dho. This indentation can beeither (A) to the same penetration depth hsmax as the indentation instep (1), which case hsmax should be obtained in step (1), or (B) to thesame load, such that Pomax=Psmax.

(3) determine R=A/Ao from

    R={dPs/dhs}.sup.2 {dPo/dho}.sup.-2                         (17)

(4) determine either (i)

    As={(dPs/dhs)/(CuE*}.sup.2                                 (18a)

or (ii)

    Ao={(dPo/dho)/(C.sub.u E*}.sup.2                           (18b)

where, as is known in the art, C_(u) is a constant that depends on thetype of the indenter (e.g., 1.142 for tetragonal vickers and 1.167 fortrigonal berkovich pyramidal indenter) and, as is known in the art,

    E*={(1-v.sup.2)/E+(1-(v.sub.in).sup.2)/E.sub.in }.sup.-1,

where v=the Poisson ratio characteristic of the material 40, v_(in) isthe Poisson ratio of the indenter 30, E is the Young's moduluscharacteristic of the material 40 and E_(in) is the Young's moduluscharacteristic of the indenter.

(5) For case (A), indentation in to the same penetration depth hs(max),determine Pave from

    Pave=Po(max)/Ao or Pave=Ps(max)/As

For case (B), indentation to the same load, Ps(max), determine Pave fromPave=Po(max)/Ao=Psmax/Ao

(6) For (A) use the equal penetration equation 12; for (B) use the equalload equation 13, to determine the stress in the stressed section 42.

In the foregoing strategy, little second data is required. The seconddata, that is, data characteristic of the material includes the Young'smodulus and the Poisson ratio. The known data of the indenter includesthe Poisson ratio, the constant C_(u), and the Young's modulus, and theangle of indentation of the indenter.

Strategy X

Proceed generally as in Strategy IX, however, rather than obtain an Echaracteristic of the material, as well as the E_(in) and the Poissonratios v and v_(in), observe with area measurement device 50 the area ofidentation of either the stressed section 42 or the substantially stressfree section to obtain either As or Ao, the calculate E* from theequation ##EQU14##

Note that the load P, indicated by reference numeral 106, measured forin the case of compressive stressing of the section 42 is larger thanthe load Pm.

Note that many of the strategies above do not require an unloadingcurve, or that all of a loading curve be recorded or noted. Whenobtaining first data that includes any two of, or all of, an area As, apenetration hs and a load Ps, it is preferable that the data he takenfor the maximum load Psmax or the maximum penetration, hsmax, forenhanced accuracy. However, as understood by one of ordinary skill inthe art, in light of the disclosure herein, it often not strictlynecessary that data be the "maximum" data.

In addition, strategies based on equation 11, such as those specificallyenumerated above, determine the stress in the stressed section 42 as ifthe stress is constant with depth of d_(R) (FIG. 4). Such adetermination may be useful in instances where the stress is in fact notconstant. The usefulness will depend on the degree to which the stressis not constant and on the accuracy required, i.e., on the applicationin which the stress data is to be used. As indicated elsewhere, oneadvantage of the present invention is that method and apparatus thereofcan be practiced using nanoindentation, microindentation, ormacroindentation. If it is known apriori that the stress in the stressedsection 42 is in fact constant to a depth d_(R), indicated by referencenumeral 73 in FIG. 4, is preferable that the indenter be selected suchthat the indenter contact size (also referred to as contact radius,where contact diameter is twice the contact radius) a, indicated byreference numeral 82 in FIG. 7, be less than d_(R) ; more preferably,the contact size a is less than approximately 1/3 of d_(R) ; mostpreferably, the contact size a is less than approximately 1/7 of d_(R).In general, the contact size a, can be defined as a=(A/pi)^(1/2), whereA is the projected contact area.

One of ordinary skill in the art, in light of the disclosure herein, canreadily envision variations of the above strategies. Such variations areconsidered within the scope. For example, the second data characteristicof the material can include the hardness Pave, which as indicated abovecan be obtained from tables. However, as understood by one of ordinaryskill in the art, the hardness can be calculated from second data thatincludes a stress strain curve of the material or Young's modulus, theyield strength and either the stress and approximately 29% plasticstrain or the strain hardening exponent of the material. The hardnesscan also be obtained from an indentation test on a section having aknown stress of the material or a section having a known stress of amaterial substantially similar to the material 40.

Furthermore, an indentation area A can be obtained from the unloadingslope dP/dh, rather than from the measurement device 50. See WO 97/39333(PCT/US97/06425), as noted above.

In many of the strategies above, a section without stress is indented toobtain second data. As understood by one of ordinary skill in the art,in light of the disclosure herein, such second data can be obtained froman indentation of a section having a known stress.

An Alternative Approach and Determining Spatially Varying Stresses

The stress in the stressed section 42 of the material 40 can vary in thez-direction from the value at the surface 76. Accordingly, the presentinvention provides methods and apparatus for determining the variationof the stress in the z direction. According to the invention, it isrecognized that an analogy exists between the problem of determiningstress in a graded material and the present problem of determiningstress with z, or equivalently, the penetration h, into the stressedsection 42.

Consider that an elastic stress, σ_(x),0^(R) (z)=σ_(y),0^(R) (z), thatvaries (either increases or decreases) either below the indented surface76, at least over a depth of d_(R). Preferably, the material 40 iselastic-perfectly plastic.

According to the invention, an analogy is drawn between a homogeneouselastic-perfectly plastic material that has spatially varying elasticstresses and a graded elastic-perfectly plastic material with nostresses whose yield strength varies as a function of depth z. Thisanalogy is schematically illustrated in FIGS. 11A and 11B.

An equi-biaxial stress that varies with depth would result in differentlevels of additional straining during indentation locally at differentdepths below the indented surface 76. This differing level of additionalstraining causes plastic yielding to occur at different times atdifferent depths during indentation loading. This process isschematically sketched in FIG. 11A, where reference numeral 120indicates a family of curves illustrating the additional straining withdepth z.

Now consider an elastic-perfectly plastic material which contains agradient in composition that results in a gradient in yield strength asillustrated in FIG. 11B. The gradient in yield strength, indicated bythe family of curves 124, is of the same functional form as the gradientin residual stress with depth indicated by reference numeral 120 in FIG.11A. That is, if σ_(x),0^(R) (z)=σ_(y),0^(R) (z) is linear with z, so isσ_(y) (z) with z. The mechanics of indentation for such gradedelastoplastic materials have recently been solved. According to thepresent invention, from the foregoing analogy between gradient residualstresses and gradient properties of residual-stress-free materials, theunknown stress fields in the material 40 are determined by indenting thestressed section 42 with the indenter 30.

Consider first the homogeneous material 40 with stresses spatiallyvarying below the indented surface. Let the most general stress statefor this case during indentation is given by the components σ_(ij')^(R).

In addition, the condition for plasticity, as per the von Mises yieldcriterion, leads to ##EQU15##

Instead of the von Mises yield criterion, one can, with appropriatemodification, use the Tresca yield criterion.

The determination of σ₃₃ ^(R) provides a complete solution for theequi-biaxial residual stress field with spatial variation.

Consider next the analogy of a graded material (without any residualstresses) with spatially varying yield strength, as illustrated in FIG.11B. Let the most general stress state for this case during indentationis given by the components σ_(ij). The condition for plasticity, as perthe von Mises yield criterion, leads in this case to

    (σ.sub.11 -σ.sub.22).sup.2 +(σ.sub.22 -σ.sub.33).sup.2 +(σ.sub.33 +σ.sub.11).sup.2 +6{(σ.sub.12).sup.2

    +(σ.sub.23).sup.2 +(σ.sub.31).sup.2 }=2(σ.sub.y.sup.g (z)).sup.2 =2[σ.sub.y.sup.2 -{σ.sub.x,0.sup.R (z)}.sup.2 ], for σ.sub.y.sup.2 >{σ.sub.x,0.sup.R (z)}.sup.2. (20)

In this equation, σ_(y) ^(g) (z) is the spatially varying yield strengthof the graded elastoplastic solid. The analogy is completed by notingthe following: ##EQU16## From the auxiliary solution involving σ_(ij)for a graded material without residual stresses, a solution for theresidual field σ_(ij) ^(R) can be obtained.

Let the variation of stress with depth in the material 40 be modeled aslinear, and hence characterized by the following equation:

    σ.sub.x,y.sup.R =σ.sub.y,0.sup.R (z)=G+Γz, (22)

where G is the magnitude of the equi-biaxial residual stress at theindented surface 76, and Γ indicates the steepness (slope) of theincrease or decrease of the residual stress with depth z. Theelastoplastic deformation of the material 40 with a linearly varyingresidual stress field has, as noted earlier, an analogy with theelastoplastic deformation of a graded material with linearly varyingyield strength (but without any residual stresses). Now applying thisanalogy, and noting the results of Equations (20) and (22), it is seenthat ##EQU17##

For reasonably large magnitudes of surface residual stresses |G| and/ormoderate variations of the residual stress steepness |Γ|, Equation (23)can be further linearized in the following manner:

    (σ.sub.y.sup.g (z)).sup.2 ≈(σ.sub.y.sup.2 -G.sup.2)-2GΓz.                                     (24)

Now consider the analogous case of the graded material with a linearvariation in yield strength of the form:

    σ.sub.y (z)=σ.sub.y,0.sup.g +bz,               (25)

where σ_(y),0^(g) is the yield strength of the graded material (withoutresidual stresses) at the indented surface and b is the steepness of thevariation of yield strength with depth z. Linearizing the square ofEquation (25),

    (σ.sub.y,0.sup.g +b,z).sup.2 ≈(σ.sub.y,0.sup.g).sup.2 +2bσ.sub.y,0.sup.g z.                               (26)

This approximation holds for large values of σ_(y),0^(g) and/or moderatevariations in yield strength steepness |b|. Matching the appropriateterms of Equations (25) and (26), there is obtained

    b=-GΓ, (σ.sub.y,0.sup.g).sup.2 =(σ.sub.y,0.sup.g -G.sup.2).                                                (27)

Now using the indentation analysis of graded materials, the averagepressure, P_(ave) and the P-h relationships for the graded material are,respectively, ##EQU18## For b=0, Γ=0 and the average pressure and P-hrelationships for the homogeneous elastic-perfectly plastic solid withuniform or no residual stress are determined. (i.e. P˜h² and P_(ave) isinvariant).

From Equation (29), we see that the P-h relation for a linear gradientin residual stress field is of the form

    P=B.sub.1 h.sup.2 +B.sub.2 h.sup.3,                        (30)

where B₁ =11.88(tan γ)² σ_(y),0^(g) and B₂ =8√3b(tan γ)³ can be obtainedfrom a least-square fit to the experimentally obtained P-h curve duringindentation loading in the presence of gradient residual stress field.From this, we solve for σ_(y),0^(g) and b, respectively. Then fromEquation (27), we obtain G and Γ, and from Equation (22), σ_(x),0^(R)(z)=G+Γz is fully solved.

FIG. 12 graphically illustrates the effect of a linear variation in theelastic residual stress field on the P-h curve indentation of thematerial with an indenter 30. P-h curve 103 represents indentation of asection of the material in which the stresses do not vary with depth.Kick's law is obeyed. P-h curves 134 and 136 are for a compressive andtensile stress at the surface of the stressed section 42, respectively.As illustrated in FIG. 12, for a constant depth of penetration hsmax ofthe indenter 30, tensile residual stresses at the surface lead to adecrease of area of indentation A; compressive residual stresses, on theother hand, cause an increase in A at fixed hsmax. Likewise, underconstant indentation load Psmax, tensile stresses lead to an increase inthe area of indentation A and compressive stress leads to an increase.This is the same as for the case of the spatially uniform stressillustrated in FIG. 10.

Note that it can be easily shown in Equation (27) that bGΓ≦0, and that Gcan be positive or negative. From this, the sign of b is automaticallyfixed. Therefore, if the sign of G is determined, the magnitude and signas well the gradient of the residual stress field is fully known, asillustrated in FIG. 12.

One of ordinary skill in the art, in light of the disclosure herein,will appreciate that for the indentation of graded elastoplasticmaterials, quadratic variations in residual stresses with depth h canalso be determined by indentation by invoking the above analogy andprocedures. Accordingly, such quadratic modeling is also possible for astress that varies with depth, and such a determination according to aquadratic model is deemed within the scope of the invention.

Exemplary strategy for determining stress and the variation of thestress with h

(1) indent the stressed section 42 of the material 40 with an indenterto obtain a loading curve of load P versus penetration h.

(2) fit the loading to curve to the polynomial expression B₁ h² +B₂ h³to obtain the constants B₁ and B₂ ;

(3) determine third and fourth constants B₃ and B₄ in satisfaction ofthe formulas

    B.sub.3 =B.sub.1 /[11.88(tan Γ).sup.2 ]              (31)

    B.sub.4 =B.sub.2 /[8√3(tan Γ).sup.3 ]         (32)

where g is the angle of indentation of the indenter.

(4) obtain a value for the yield strength (σy) of the material. Thisvalue can be obtained from a tabulated source, from any of many testknown in the art, such as a bending, tension compression, and includingan indentation test.

(5) determining at least one of the magnitude of the stress at thesurface of the section of material, G, and the magnitude of the rate ofchange of the stress with penetration G, in satisfaction of the formulas

    B.sub.3 =-GΓ                                         (33)

and

    B.sub.4 =(σ.sub.y).sup.2 -G.sup.2                    (34)

(6) determine the sign of G and of Γ so as to determine whether thestress at the surface is compressive or tensile and whether it decreasesor increases with penetration. Several techniques can be employed todetermine the sign of G and G

(a) indent a section having a known stress of the material or sectionhaving a known stress of a material substantially similar to thematerial, or

(b) obtain a stress-strain curve characteristic of the material; or

(c) obtain values characteristic of the material for Young's modulus andone of the strain hardening exponent and the stress at approximately 29%plastic strain.

Note that the foregoing exemplary strategy can be used to determine thestress at the surface 76 of the stressed section, and thus, if theassumption that the stress in constant with h can be tolerated, canprovide an alternative method to those above for determining the stressin the stressed section. Preferably, when using foregoing strategy, thematerial remains elastic.

An Approach for Determining Plastic Strain

According to one aspect of the invention, methods and apparatus areprovided for determining the effective plastic strain when the stressedsection 42 has plastically yielded. If an elastoplastic bulk material oran elastoplastic thin film on a substrate contains equibiaxial residualplastic strains, the magnitude of such plastic strains can also often bedetermined by indentation.

Returning to FIG. 3, the problem of determining plastic strain,indicated by reference numeral 150, is graphically illustrated. Based ona knowledge of the stress strain curve 148 and the residual plasticstress, indicated by reference numeral 152, the total strain, indicatedby reference numeral 154, can be determined. Using the Young's modulusE, which characterizes the linear portion 156 of the stress strain curve148, the effective plastic strain 150 can be obtained from the totalstrain 154.

Consider an isotropic bulk material or thin film whose elasticproperties (i.e. E and v) and elastoplastic stress-strain response(e.g., a uniaxial stress/strain curve) are known apriori.

Conditions governing the invariance of hardness (contact pressure)during elastoplastic indentation with or without pre-existing plasticstrain are described in the Appendix.

When the Young's modulus, E, and the Poisson ratio, ν, of the materialare different from the corresponding values E_(in) and ν_(in)respectively, of the indenter, the general expression for theindentation modulus is given by ##EQU19## From the initial part of theunloading curve, the contact area A at the maximum load P_(max) for thesubstrate with the residual stress is given by (Giannakopoulos et al.,1994; Suresh et al., 1996): ##EQU20## where dP/dh is the slope of theinitial unloading portion of the P-h curve for the substrate with theresidual stress, C_(u) is a non-dimensional constant which depends onlyon the shape of the indenter (C_(u) =1.142 for the tetragonal Vickerspyramid indenter and C_(u) =1.167 for the trigonal Berkovich pyramidindenter). Similarly, for the surface without any residual stress,calculate, from a knowledge of the elastic modulus of the bulk of thethin-film material, the corresponding area of contact for the sameapplied load P₀ : ##EQU21##

If it is possible to perform an indentation in a stress-free section ofthe material, measure A₀ from that indentation experiment and comparewith that predicted from Equation (37).

It has been shown through numerical simulations of sharp indentation ofelastoplastic materials (Giannakopoulus et al., 1994) that ##EQU22##where σ_(u) is the yield stress at a plastic strain of approximately0.3, and σ_(y) ^(R) is the yield strength of the plastically prestrainedmaterial (yield strength of the material with residual strains). Asimilar expression A₀ to σ_(y), σ_(u) and α. The contact area rationthen can simply be written as ##EQU23## Equation (21) can be solvednumerically for σ_(y) ^(R), using the known area ratio, A/A₀, atconstant load, P₀. (The other material constants are assumed to be knownor measured independently from other mechanical tests). The region ofvalidity of this analysis is: σ_(y) ≦σ_(y) ^(R) ≦σu. Clearly, in theabsence of residual strains, A/A₀ →1, Equation (21) gives σ_(y) ^(R)→σ_(y), as expected. Once σ_(y) ^(R) is so determined, the knownstress-strain curve for the material can be used to readily findε_(x),0^(p).spsp.1 =2|ε_(x),0^(p).spsp.1 |=2|ε_(y),0^(p).spsp.1 |. Themagnitude of the equibiaxia residual strain 2|ε_(x),0^(p).spsp.1|=2|ε_(y),0^(p).spsp.1 | is thus determined.

The predictions of the average contact pressure based on the analyticaltheory, for indented surfaces with pre-existing residual plasticstrains, are compared above in Tables I and II with finite elementsimulations. These two tables provide a comparison of the analytical andcomputational predictions for indented materials without and with strainhardening, respectively, identified through indentation when thematerial exhibits strain hardening. For both elastic-perfectly plasticand isotropically strain hardening materials with pre-existing residualplastic strains, the finite element results corroborate the trendspredicted by the analysis.

Exemplary Strategies for Determination of Effective Plastic Strain

Strategy I

(1) Indent the stressed section 42 of the substrate 40 to obtain an areaAs.sub.(max) at a load Ps.sub.(max)

(2) Indent a substantially stress free section of the material 40 or ofa material susbtantially similar thereto to obtain an area Ao at a loadPo substantially equal to Ps.sub.(max)

(3) Obtain second data characteristic of the material, namely, yieldstrength, stress at approx 30% strain and Young's modulus E.

(4) determine the residual yield strength σ_(y) ^(R) from equation (2)(39).

(5) with the knowledge of σ_(y), σ_(y) ^(R), E, and σ_(u) determine thetotal strain corresponding to σ_(y) ^(R), and determine from E theplastic strain, as illustrated in FIG. 13.

Strategy II

(1) Indent the stressed section 42 of the material 40 a sharp indenterto obtain a loading and unloading curves of load P verses penetration h.Obtain h_(r),P_(max),A_(max),dP/dh. Amax is to be measured by areameasurement device 50.

(2) From the unloading slope and Amax determine E* and E ##EQU24##C*=1.142 (Vickers), 1.167 (Berkovich). ##EQU25## (2.1) If P_(max)/A_(max) E tan a) .English Pound.0.1 ##EQU26##

If P_(max) /(A_(max) E tan a)>0.1, then proceed with the remainder ofthe steps

(3) Solve A and B, simultaneously and obtain σ_(u) and σ_(y) ^(R)##EQU27##

(4) Solve for H₀ and then for σ_(y) from (C) and (D) ##EQU28##

    σ(D) .sub.y =σ.sub.u -0.29H.sub.0

(6) Obtain the strain hardening, η ##EQU29##

(7) with the knowledge of σ_(y), σ_(y) ^(R), η, E, and σ_(u), or h, anddetermine the total strain corresponding to σ_(y) ^(R), and determinefrom E the plastic strain, as illustrated in FIG. 3.

Strategy III

(1) Indent the stressed section 42 of the material 40 with a sharpindenter to obtain loading data of Psmarx and a corresponding hsmax.

(2) obtain second data characteristic of the material-E, σ_(u), σ_(y)

(3) determine σ_(y) ^(R) from equation (B) of strategy II

(4) with the knowledge of σ_(y), σ_(y) ^(R), E, and σ_(u), determine thetotal strain corresponding to σ_(y) ^(R), and determine from E theplastic strain, as illustrated in FIG. 3.

Strategy IV

(1) Indent the stressed section 42 of the material 40 with a sharpindenter to obtain loading data of hs.sub.(max) and hr.

(2) obtain second data characteristic of the material-E, σ_(u), andσ_(y)

(3) determine σ_(y) ^(R) from equation (A) of strategy II

(4) with knowledge of σ_(y), σ_(y) ^(R), E, and σ_(u), determine thetotal strain corresponding to σ_(y) ^(R), and determine from E theeffective plastic strain, as illustrated in FIG. 3.

Strategy V

(1) Indent the stressed section 42 of the material 40 a sharp indenterto obtain a loading and unloading curves of load P verses penetration h.Obtain h_(r), Pmax, hmax, hr and dP/dh.

(2) Obtain second data characteristic of the material, namely, E

(3) Use the equations in step (2) of Strategy II to determine Amax

(4) Proceed as in Strategy II starting with step (3)

The methods disclosed herein are suitable for programming on andsolution by an appropriate data processor. Illustrated in FIG. 13 isapparatus of the present invention that can allow determination of thestresses in the stressed section 42 of the material 40 of FIG. 1. Acontrol panel 200 can be used to manually control the loading frame 22,which mounts the indentation apparatus 20, and the load P applied to theindenter 30. A fast D/A converter 204 is connected via lead 208 to thedisplacement sensor 46, via lead 208 to load cell 24, and via lead 210to the area measurement device 50, if present, so as to acquire load anddepth of penetration measurements via a computer 214. LABVIEW softwarecan be used for data acquisition. The computer 214 which can directlyacquires data via the D/A converter and can control the test by standardgeneral purpose interface bus (GPIB) connections 216. The computer 214also can be arranged to control an X-Y stage (not shown) upon which thematerial 40 can be mounted, and other adjustable components such asmirror 48 so that an entire set-up and testing procedure is essentiallyautomated, and indentations can be performed at pre-selected locationsserially, such as the stressed section 42 and a section having a knownstress.

An example of a suitable data processor 215 is illustrated in FIG. 14.FIG. 14 represents the computer 214, or a calculator, or dedicatedintegrated circuit. The data processor includes a central processor 220connected to a memory 222 via an interconnection mechanism 92. An inputdevice 226 is also connected to the processor and memory system via theinterconnection mechanism, as is an output device 228.

It should be understood that one or more output devices 228 may beconnected to the computer 214. Example output devices include cathoderay tube (CRT) displays, liquid crystal displays (LCD), printers,additional storage devices and control outputs via the GPIB connections216 and communication devices such as a modem. It should also beunderstood that one or more input devices 226 may be connected to thecomputer 214. Example input devices include GPIB connections 216, akeyboard, keypad, track ball, mouse, pen and tablet and communicationdevice. It should be understood the invention is not limited to theparticular input or output devices used in combination with the computer214 or to those described herein.

The computer 214 may be a general purpose computer system which isprogrammable using a high-level computer programming language, such as"C," "Pascal" or "Visual Basic". The computer may also be speciallyprogrammed, using special purpose hardware. Additionally, the computer214 may be a multiprocessor computer system or may include multiplecomputers connected over a computer network.

In a general purpose computer system, the central processor 220 istypically a commercially available central processor, of which theseries x86 processors, available from Intel, and the 2040X0 seriesmicroprocessors available from Motorola are examples. Many otherprocessors are available. Such a microprocessor executes a programcalled an operating system, of which UNIX, DOS and VMS are examples,which controls the execution of other computer programs and providesscheduling, debugging, input/output control such as for the GPIBconnections, accounting, compilation, storage assignment, data andmemory management, communication control and related services. Thecentral processor and operating system define a computer platform forwhich application programs in various programming languages may bewritten. It should be understood the invention is not limited to aparticular computer platform, particular data processor 215, orparticular high-level programming language.

An example memory system 222 will now be described in more detail inconnection with FIG. 15. A memory system typically includes a computerreadable and writeable nonvolatile recording medium 232, of which amagnetic disk and tape are examples. The disk may be removable, known asa floppy disk, or permanent, known as a hard drive. In the embodimentillustrated in FIG. 16, the medium 232 is a disk, which is shown in FIG.16 has a number of tracks, as indicated at 234, in which signals arestored, typically in binary form, i.e., a form interpreted as a sequenceof one and zeros such as shown at 236. Such signals may define anapplication program to be executed by the microprocessor, or informationstored on the disk to be processed by the application program.Typically, in operation, the central processor 220 causes data t be readfrom the nonvolatile recording medium 232 into an integrated circuitmemory element 238, which is typically a volatile, random access memorysuch as a dynamic random access memory (DRAM) or static memory (SRAM).The integrated circuit memory element 238 allows for faster access tothe information by the processor than does the medium 232. The processorgenerally manipulates the data within the integrated memory circuitmemory 238 and then copies the data to the medium 232 when processing iscompleted. A variety of mechanisms are known for managing data movementbetween the medium 232 and the integrated circuit memory element 238,and the invention is not limited thereto. It should also be understoodthat the invention is not limited to a particular memory system 222.

FIG. 16 is a high level block diagram indication one manner in which thedata processor can proceed to the determine the stresses and/or strainaccording to the invention. One of ordinary skill in the art, in lightof the disclosure herein, can readily program a computer system tofollow a particular approach and strategy to determine appropriateoutput data, such as the stress in the stressed section 42 of thematerial 40 of FIG. 1. In box 302 a particular strategy is selected,such as for example, the determination of stress in the stressed section42 using indentation tests of the stressed and an unstressed section ofthe material 40. Proceeding to box 304, the first data and second dataare obtained, such as by the computer 214 controlling the load frame andindentation apparatus to indent the stressed and unstressed sections ofthe material. Next, in box 306, the stress or other determination ismade according to the procedures of the particular strategy andapproach, many examples of which are given above. Finally, thedetermined quantity, such as the stress or variation thereof withpenetration, is displayed or stored in memory. Note that if the dataprocessor 215 is processing data files of preobtained first data, therelationship of box 302 and 304 can be reversed. That is, the strategyand approach to follow can be determined in whole or in part by thepreobtained first and second data.

As is seen from the foregoing, the invention advantageously providesmethod and apparatus for determining, from a simple indentation test ofa stressed section of a material, the stress in the section, thevariation in stress with depth, and the plastic strain in the sectionand residual yield strength. First data are obtained from an indentationtest. Second data are data that characterize the materail and that arenecessary, given the first data to be used in a particular strategy andapproach, to obtain a solution according to that strategy and approach.For example, in one strategy second data may only need to include theYoung's modulus characteristic of the material. Indentation is performedwith an indenter, and certain properties of the indenter can enter intoa solution, such as the angle of indentation, given the approach, thestrategy, and the first and second data to be used in the determinationof the stress, variation thereof, and/or the strain.

The invention can be practiced according to at least three generalapproaches and a number of strategies. In particular, the strategiesdiscussed in detail in the first approach wherein the stress in thestressed section is determined as if constant are intended to serve asexemplary of the variations possible in the other strategies outlined.All strategies that follow one of the approaches disclosed herein aredeemed to be within the scope of the invention.

The invention is deemed to include apparatus, such as data processor,such as a computer, which includes program elements stored in a memoryfor making determinations according to the invention. The computersystem can obtain first data directly from the indentation apparatus orcan process files including such data. The invention is also deemed toinclude an article of manufacture, such computer data product (e.g., afloppy diskette) modified to including program elements in a binary orother representation for determining stresses, strain and otherquantities according to the techniques disclosed herein.

As noted above, the invention advantageously does not appear to belimited to any particular scale. The invention can be practiced todetermine the stresses in large structures, such asrolled/extruded/forged plates, shafts subjected to surface treatments,coated, shot peened or laser shock peened components, case hardenedmaterials, or rapidly quenched materials. Testing of such largercomponents can be accomplished by placing the object in a mechanicaltesting machine, such as a suitably scaled indentation apparatus 20, anprobing the component with, for example, a cone. Such testing could beapplied in the environment of the production floor as a quality controlmeasure in heavy industries.

Finally, the first data are obtained from an indentation test of thestressed section of material. Preferably, the indenter used in theindentation test is a sharp indenter. With reference to FIG. 7B, sharp,as used herein, means that preferably the projected contact diameter 2a,where a is indicated by reference numeral 82 in FIG. 7B of the indenteris at least three (3) times the tip radius of the indenter; morepreferably, the contact diameter 2a is at least six (6) times thatcontact diameter of the indenter; and, most preferably, the contactdiameter 2a is at least ten (10) time the tip radius for the indenter.Accordingly the invention is deemed to be useful with indenters that canbe characterized as "blunt" tipped indenters

We claim:
 1. A method of determining the preexisting stress in astressed section of a material, comprising:obtaining first data from theindentation of the stressed section of the material with a sharpindenter, the first data including at least two of a load on theindenter P_(s), an area of indentation A_(s) and a depth of penetrationh_(s), obtaining second data including at least one of 1) data obtainedfrom an indentation of a second section having a known stress of one ofthe material and a second material substantially similar to the materialand 2) additional data that can allow determination of the Young'smodulus that is characteristic of the material, and determining from thefirst and second data the stress in the stressed section of material. 2.The method of claim 1 wherein obtaining first data includes obtainingthe depth of penetration h_(s).
 3. The method of claim 1 whereinobtaining first data includes obtaining the area As from unloading theindenter and determining the initial unloading slope of load withpenetration dP/dh.
 4. The method of claim 1 wherein obtaining first dataincludes determining the area of indentation A_(s) by observation. 5.The method of claim 1 wherein obtaining the first data includesobtaining the area of indentation A_(s).
 6. The method of claim 1wherein obtaining the first data includes obtaining P_(s).
 7. The methodof claim 1 wherein obtaining the first data includes obtaining onlyA_(s) and P_(s).
 8. A method of determining the preexisting stress in astressed section of a material, comprising:obtaining indentation datafrom first and second indentations of substantially the same indentationload, the first indentation being of the stressed section of thematerial and the second indentation being of a second section having aknown stress of one of the material and a substantially similarmaterial, the indentation data including at least a first depth ofindentation of the first indentation and a second depth of indentationof the second indentation; obtaining additional data including one of:a) the indentation load and the area of the second indentation; and b)other data that can allow the determination of the hardness that ischaracteristic of the material; and determining the stress from thedata.
 9. The method of claim 1 including the step of selecting theindenter to be an indenter of a type selected from the group consistingof a Vickers indenter, a conical indenter and a Berkovich indenter. 10.The method of claim 1 wherein the stresses in the stressed section ofmaterial are substantially uniform over a predetermined depth, andincluding the step of selecting the indenter to have a contact size lessthan approximately one-third of the predetermined depth.
 11. The methodof claim 1 wherein the stresses in the stressed section of material aresubstantially uniform over a predetermined depth, and including the stepof selecting the indenter to have a contact size less than approximatelyone-seventh of the predetermined depth.
 12. The method of claim 1wherein obtaining the second data includes obtaining said additionaldata including a stress-strain relationship characteristic of thematerial.
 13. The method of claim 1 wherein obtaining second dataincludes obtaining the following: the yield strength (σy), one of thestrain hardening exponent and the stress at approximately 29% plasticstrain and said additional data including the Young's modulus.
 14. Themethod of claim 1 wherein obtaining the second data includes obtainingdata from the indentation of the second section.
 15. The method of claim1 wherein obtaining the second data includes obtaining data from theindentation of the second section, the second section beingsubstantially without stress when indented.
 16. The method of claim 1wherein obtaining the second data includes obtaining data from theindentation of the second section of the second material.
 17. The methodof claim 1 wherein obtaining the second data includes obtaining datafrom the indentation of the second section, the second section beingsubstantially without stress and being of the second material.
 18. Themethod of claim 1 wherein obtaining the second data includes obtaining avalue for the hardness of the material.
 19. The method of claim 1wherein obtaining the second data includes obtaining data from theindentation of the second section to a load P_(o) substantially equal toa selected load on the indenter when indenting the stressed section toobtain the first data.
 20. The method of claim 1 wherein obtaining thesecond data includes obtaining data from the indentation of the secondsection to a penetration substantially equal to a selected penetrationof the indenter into the stressed section when indenting the stressedsection to obtain the first data.
 21. The method of claim 1 whereinobtaining the first data includes obtaining the area of indentationA_(s) and wherein obtaining the second data includes obtaining saidadditional data including an unloading slope of load with penetration.22. The method of claim 1 wherein obtaining the first data includesobtaining the depth of penetration h_(s) and wherein obtaining thesecond data includes obtaining from the indentation of the secondsection an area A_(o) corresponding to a load substantially equal to theload on the indenter when indenting the stressed section to thepenetration h_(s), the second section being substantially without stresswhen indented.
 23. The method of claim 1 wherein obtaining the firstdata includes obtaining the load P_(s) and wherein obtaining the seconddata includes obtaining an area A_(o) and a load P_(o) from theindentation of the second section to a penetration substantially equalto the penetration into the stressed section corresponding to the loadP_(s), the second section being substantially without stress.
 24. Themethod of claim 22 wherein obtaining the second data includes obtainingthe second area of indentation from an observation thereof.
 25. Themethod of claim 22 wherein obtaining the second data from theindentation of the second section includes loading and unloading toobtain the slope of load with penetration upon initial unloading dP_(o)/dh to obtain the second area of indentation therefrom.
 26. The methodof claim 22 wherein obtaining the first data includes loading andunloading to obtain the slope of load with penetration d P_(s) /dh uponinitial unloading;wherein obtaining the second data includes obtainingthe second data from the indentation of the second section includingloading and unloading to obtain the slope of load with penetration uponinitial unloading dP_(o) /dh; and wherein determining the stressincludes taking a ratio involving dP_(o) /dh and dP_(s) /dh.
 27. Themethod of claim 1 wherein obtaining first data includes obtaining datafrom the indention including loading and at least initial unloading ofthe indenter such that the first data includes a first initial unloadingslope of load with penetration, dPs/dhs;wherein obtaining the seconddata includes obtaining data from the indentation of the second sectionto obtain second data including a second initial unloading slope of loadwith penetration, dPo/dho; and wherein determining the stress includesdetermining the square of the ratio of the first unloading slope to thesecond unloading slope, {(dP/dhs)/(dPo/dho)}².
 28. The method of claim 1wherein obtaining the first data includes obtaining the area ofindentation by observation of said area; andwherein obtaining the seconddata includes obtaining said additional data including the initialunloading slope of load to penetration of the indentation of thestressed section.
 29. The method of claim 1 wherein determining thestress includes determining the stress according to one of an equal loadstrategy and an equal penetration strategy.
 30. The method of claim 1wherein the step of determining the stress includes determining a ratioequivalent to a ratio of indentation areas and determining a hardnessthat is characteristic of the material.
 31. The method of claim 1wherein determining the stress includes determining the stress inaccordance with an equation of the form

    R={1+(stress/Pave)geomf}.sup.-1

where R is a ratio equivalent to the ratio of the indented area As ofthe section to an indented area Ao characteristic of the materialwithout stress, Pave is the hardness that is characteristic of thematerial and is determined from at least one of the first and seconddata, and geomf is an optional geometric factor dependent on the shapeof the indenter.
 32. The method of claim 31 wherein the stress isdetermined according to an equal penetration depth strategy by using theequation as follows:

    if R>1, then R={1-(sin α)stress/P.sub.ave }.sup.-1

    if R<1, then R={1+stress/P.sub.ave }.sup.-1

where α is an angle related to the angle of indentation of the indenter.33. The method of claim 31 wherein the stress is determined according toan equal load strategy by using the equation as follows:

    if R>1, then R={1-stress/P.sub.ave }.sup.-1

    if R<1, then R={1+(sin α stress/P.sub.ave }.sup.-1

where α is an angle related to the angle of indentation of the indenter.34. A method of determining the preexisting stress in a stressed sectionof a test material using first and second indentations, the testmaterial being one of an elastoplastic material and/or a materialexhibiting strain hardening, comprising:obtaining indentation data fromfirst and second indentations with at least one sharp indenter, thefirst indentation being of the stressed section of the test material andthe second indentation being of a second section having a known stressof one of the test material and a substantially similar material, theindentation data including at least the area of indentation of the firstindentation and the area of indentation of the second indentation;obtaining additional data including one the following: an indentationload of one of the indentations; and other data that can allow thedetermination of the hardness that is characteristic of the material;and determining the stress from the data.
 35. A method of determiningthe stress in a stressed section of a material, comprising:(a) obtainingfirst data from the indentation of the stressed section with anindenter, the first data including at least a load P_(s) and a depth ofpenetration h_(s) ; (b) obtaining second data characteristic of thematerial, the second data including Young's modulus (E), the yieldstrength (σy), and one of the stress at approximately 29% plastic strain(σu) and the strain hardening exponent (η); (c) determining the hardnessP_(ave) ; (d) determining (h_(o))² =P_(s) {σy(1+σu/σy)C*(1+ln((tanα)E/3σ_(y)))}⁻¹ where C* and tan α depend on the type of the indenter;(e) determining the ratio R equivalent to R=(h_(s))² /(h_(o))² ; (f)determining the stress in satisfaction of the following formulae;

    if R<1; R={1+(geomf)stress/P.sub.ave }.sup.-1

    if R>1; R={1-stress/P.sub.ave }.sup.-1

where geomf=sin α, and α is related to the angle of indentation of theindenter.
 36. A method of determining the stress in a stressed sectionof a material, comprising:(a) obtaining first data from the indentationof the stressed section with an indenter, the first data including atleast a penetration h_(s) ; (b) obtaining second data characteristic ofthe material, the second data including Young's modulus (E), the yieldstrength (σy), and one of the stress at approximately 29% plastic strain(σu) and the strain hardening exponent(h); (c) determining the hardnessP_(ave), where determining the hardness includes one of1) obtainingfirst data including the load P_(s) and the corresponding indentationarea A_(s) and determining the hardness P_(ave) =P_(s) /A_(s) ; and 2)otherwise obtaining the known hardness value for the material; (d)determining (P_(o)) from P_(o) =(h_(s))² σy(1+σu/σy)C*{1+ln((tanα)E/3σy) } where C* and tan α depend on the type of the indenter e)determining the ratio R, where determining R includes one of1)determining A_(o) from A_(o) =P_(o) /P_(ave) and determining R=A_(s)/A₀, where the first data includes the area of indentation A_(s) and; 2)determining R=P_(s) /P₀, where the first data includes the load ofindentation P_(s) (f) determining the stress in satisfaction one of thefollowing formulae

    if R<1; R={1+stress/P.sub.ave }.sup.-1

    if R>1; R={1-(geomf)stress/P.sub.ave }.sup.-1

where geomf=sin(α), and α is related to the angle of indentation of theindenter.
 37. A method of determining the stress in a stressed sectionof a material, comprising:(a) obtaining a loading curve of load P versespenetration h from the indentation with a known indenter of the stressedsection: (b) fitting the loading curve to a polynomial expression of theform B₁ h² +B₂ h³ to determine first and second constants B₁ and B₂ ;(c) based on the known properties of the indenter determining at leastone additional constant B₃ from B₁ ; (d) obtaining a value for the yieldstrength (σy) characteristic of the material; (e) determining the stressG in the section of material as a function of the at least oneadditional constant B₃ and σy.
 38. The method of claim 37 whereinobtaining a loading curve includes obtaining a loading curve from theindentation by sharp indenter.
 39. The method of claim 37 inlcuding thestep of selecting the type of the indenter to be one of a Vickers,Berkovich and conical indenter.
 40. The method of claim 37 whereindetermining additional constants includes determining a constant B₃

    B.sub.3 =B.sub.1 /(K.sub.1 (tan γ).sup.2),

where K₁ is a constant.
 41. The method of claim 37 wherein determiningthe stress G includes determining the stress such that formula (G)²=(σy)² -(B₃)² is satisfied.
 42. The method of claim 37 includingdetermining whether the stress in the section of the material iscompressive or tensile.
 43. The method of claim 42 wherein determiningwhether the stress is tensile or compressive includes obtaining datafrom the indentation of one of a section having a known stress of thematerial and a section having a known stress of a second materialsubstantially similar to the material.
 44. The method of claim 42wherein determining whether the stress is tensile or compressiveincludes obtaining a stress strain curve characteristic of the material.45. The method of claim 42 wherein determining whether the stress istensile or compressive includes obtaining a value for the hardness thatis characteristic of the material.
 46. The method of claim 37 includingdetermining the change of stress with depth into the material.
 47. Themethod of claim 46 wherein determining the change of stress with depthincludes determining a fourth constant B₄ in satisfaction of the formulaB₄ =B₂ /(K2(tan γ)³) and determining the magnitude of the slope of thestress in satisfaction of the formula Γ=-B₄ /G.
 48. The method of claim46 wherein determining the change of stress with depth includesdetermining the third constant in satisfaction of the formula B₃ =B₁/(K₁ (tan γ)²);determining a fourth additional constant B₄ insatisfaction of the formula B₄ =B₂ /(K₂ (tan γ)³), where K₁ and K₂ areconstants; determining the stress G in satisfaction of the formula (G)²=(σy)² -(B₃)² ; determining the sign of G; and determining G, the slopeof change of the stress G with penetration, in satisfaction of theformula Γ=-B₄ /G.
 49. A method of determining stress at the surface of astressed section of a material and of determining the variation ofstress with penetration, comprising:(a) obtaining a loading curve ofload P versus position h from the indentation with an indenter of thesection of material; (b) fitting the loading to curve to the polynomialexpression B₁ h² +B₂ h³ ; to obtain the constants B₁ and B₂ ; (c)determining third and fourth constants B₃ and B₄ in satisfaction of theformulas

    B.sub.3 =B.sub.1 /[11.88(tan γ).sup.2 ]

    B.sub.4 =B.sub.2 /[8√3(tan γ).sup.3 ]

where g is a known angle of the indenter, (d) obtaining a value for theyield strength (sy) of the material; (e) determining at least one of themagnitude of the stress at the surface of the section of material, G,and the magnitude of the rate of change of the stress with penetration,G, in satisfaction of the formulas

    B.sub.3 =-GΓand (B.sub.4)=(σy).sup.2 -G.sup.2.


50. 50. The method of claim 49 including determining the sign of G so asto determine whether the stress at the surface of the material iscompressive or tensile.
 51. The method of claim 49 including determiningthe sign of G, thereby determining whether the stress increases ordecreases with depth into the material.
 52. The method of claim 51wherein determining the sign of G includes obtaining data from theindentation with an indenter of one of a section having a known stressof the material and a section having a known stress of a materialsubstantially similar to the material.
 53. The method of claim 51wherein determining the sign of G includes obtaining a stress-straincurve characteristic of the material.
 54. A method of determining theresidual plastic strain in a plastically-strained section of a material,comprising:(a) obtaining first data from the indentation of the sectionwith an indenter, the first data including a penetration of (h) and aload (P) on the indenter; (b) obtaining second data characteristic ofthe material; (c) determining the residual yield strength as a functionof the second data and the first data; and (d) determining the plasticstrain from the residual yield strength.
 55. The method of claim 54wherein obtaining the second data includes obtaining the Young's modulusof the material and one of the stress at approximately 29% plasticstrain and both of the strain hardening exponent and the yield strength.56. The method of claim 54 wherein obtaining the second data includesobtaining a stress-strain curve.
 57. The method of claim 54 whereinobtaining second data includes obtaining data from the indentation witha second indenter one of a section having a known stress of the materialand a section having a known stress of a material substantially similarto the material.
 58. The method of claim 54 wherein obtaining first datafrom indentation includes obtaining first data including a depth ofpenetration hs at a load P_(s) ; andwherein obtaining second dataincludes obtaining the Young's modulus (E) and one of the strainhardening exponent (h) and the stress at approximately 29% plasticstrain of the material, (σ_(u)); and wherein determining the residualyield strength includes determining the residual yield strength based onthe Young's modulus, (E) the geometry of the indenter, P_(s) and hs. 59.The method of claim 58 wherein determining the residual yield strengthincludes determining the residual yield strength such that it satisfiesthe following equation: ##EQU30## where c* is a constant that isdetermined by the type of the indenter and a related to the angle ofindentation of the indenter, and σ_(y) ^(R) is the residual yieldstrength.
 60. The method of claim 54 wherein obtaining first dataincludes obtaining first data from the indentation that includesunloading and loading such that the first data includes a maximum loadPm, a maximum depth of penetration hm and residual depth of penetrationhr;wherein obtaining second data including obtaining the Young's modulusand one of the strain hardening exponent and the stress at approximately29% plastic strain of the material; and wherein determining the residualyield strength includes determining the residual yield strength based onthe Young's modulus, hr, hm, the geometry of the indenter and one of thestress at approximately 29% plastic strain and both of the strainhardening exponent and the yield strength.
 61. The method of claim 60wherein determining the residual yield strength includes determining theresidual yield strength such that it satisfies the following equation:##EQU31## where σ_(y) ^(R) is the residual yield strength.
 62. Themethod of claim 54 wherein obtaining the first data from indentationincludes obtaining first data from the indentation including unloadingand loading such that the first data includes the slope upon initialunloading dP/dh;wherein obtaining second data includes obtaining Young'smodulus (E), and including determining the area Am corresponding to themaximum load on the indenter as a function of E and dP/dh.
 63. Themethod of claim 54 wherein obtaining the first data from indentationincludes obtaining the first data from the indentation includingunloading and loading such that the first data includes the slope uponinitial unloading, dP/dh, the first data also including the area Amcorresponding to the maximum load on the indenter; andwherein obtainingsecond data includes obtaining the Poisson ratio the material andobtaining Young's modulus of the material as a function of at leastdP/dh, Am, the Poisson ratio and the Young's modulus of the indenter.64. The method of the claim 54 wherein obtaining the first data fromindentation include obtaining first data from the indentation includingloading and unloading and observing the area of indentation to determinefirst data including a maximum load Pm, a maximum penetration hm, amaximum area of indentation Am the slope upon initial unloadingdP/dh;wherein obtaining second data includes obtaining the Poisson ratioof the material and the Young's modulus of the material as a function ofAm and dP/dh of the first data of the Poisson ratio of the indenter andof the Young's modulus of the indenter; and wherein determining theresidual yield strength includes determining the residual yield strengthbased on Young's modulus of the material, hr, hm, and Pm, an anglerelated to the angle of indentation of the indenter, and a constantdetermined by the type of the indenter.
 65. The method of claim 64wherein determining the residual plastic strain includes determining theyield strength sy in satisfaction of the following two equations:

    A.sub.m /(h.sub.m)2=9.96-12.64(1-Ho/E)+105.42(1-Ho/E).sup.2 -229.57(1-Ho/E).sup.3 +157.67(1-Ho/E).sup.4               ( 1)

    σy=(σu)-0.29Ho                                 (2).


66. The method of claim 65 wherein the residual yield strength and thestress at approximately 29% plastic strain are determined by solving thefollowing two equations: ##EQU32## where σ_(y) ^(R) is the residualyield strength, a is an angle related to the angle of indentation of theindenter, and c* is a constant related to the type of the indenter. 67.The method of claim 54 wherein obtaining first data from indentationincludes obtaining first data from the indentation including loading andunloading such that the first data includes a maximum load P_(m), amaximum penetration h_(m) and a residual penetration h_(r), whereinobtaining the second data includes obtaining the Young's modules (E) ofthe material, wherein determining the residual yield strength includesdetermining the yield strength as a function of Pm, hm, (E), hr, and anangle of the indenter (alpha) in satisfactions of the following twoequations: ##EQU33## where σ_(y) ^(R) is the residual yield strength, c*is a constant dependent or the type of the indent, and a is an anglerelated to the angle of indentation of the indenter.
 68. The method ofclaim 54 wherein obtaining first data includes obtaining an area ofindentation A_(ma) corresponding to the maximum load P_(m) on theindent;wherein obtaining the second data includes obtaining the Young'smodules (E); and wherein determining the residual yield strengthincludes evaluating the ratio ##EQU34## where a is an angle related tothe angle of indentation of the indenter, and estimating the residualyield strength to be approximately equal to ##EQU35## where the ratio isless than or equal to approximately 0.1.
 69. An apparatus fordetermining the preexisting stress in a stressed section of material,comprising:a data processor, said data processor including program meansfor determining the stress in a stressed section of a material fromfirst data and from second data, said first data obtained from anindentation test on the stressed section and including at least two of aload on the indenter P_(s), an area of indentation A_(s) and a depth ofpenetration h_(s), said second data including at least one of 1) dataobtained from an indentation of a second section having a known stressof one of the material and a second material substantially similar tothe material and 2) additional data that can allow the determination ofthe Young's modulus that is characteristic of the material.
 70. Theapparatus of claim 69 further including an indentation apparatus forindenting the stressed section of material with an known indenter fordetermining said first data.
 71. The apparatus of claim 69 wherein saidmeans for determining said stress includes means for determining saidstress from first data including a depth of penetration h_(s).
 72. Theapparatus of claim 69 wherein said means for determining stress includesmeans for determining stress from first data including an unloadingslope of load with penetration dP/dh.
 73. The apparatus of claim 69wherein said means for determining the stress includes means fordetermining the stress from second data that includes said additionaldata, said additional data including a stress-strain relationshipcharacteristic of the material.
 74. The apparatus of claim 69 whereinsaid means for determining stress includes means for determining stressfrom the second data including the additional data, where saidadditional data includes the Young's modulus, the yield strength (σy),and one of the strain hardening exponent and the stress at approximately29% plastic strain.
 75. The apparatus of claim 69 wherein said means fordetermining stress includes means for determining stress from first dataincluding an unloading slope of load with penetration, dPs/dhs and fromsecond data including a second initial unloading slope of load withpenetration, dPo/dho obtained from the indentation of the secondsection; andwherein said means determining the stress further includesmeans for determining the square of the ratio of the first unloadingslope to the second unloading slope, {(dP/dhs)/(dPo/dho)}².
 76. Theapparatus of claim 69 wherein said means for determining stress includesmeans for determining stress from the second data including theadditional data including at least one of an unloading slope dP/dh andthe area of indentation A_(s) of the stressed section.
 77. The apparatusof claim 69 wherein said means for determining stress includes means fordetermining a ratio equivalent to a ratio of indentation areas and meansfor determining a hardness that is characteristic of the material. 78.The apparatus of claim 69 wherein said means for determining the stressfrom said first and second data includes means for determining thestress in accordance with an equation of the form

    R={1±(stress/P.sub.ave)geomf}.sup.-1

where R is a ratio equivalent to the ratio of the indented area A_(s) ofthe section to an indented area characteristic of the material withoutstress, P_(ave) is the hardness that is characteristic of the materialand is determined from at least one of the first and second data, andgeomf is a geometric factor that can optionally be included and thatdepends on the indenter.
 79. The apparatus of claim 69 wherein saidmeans for determining the stress includes means for determining stressfrom first data including the load P_(s) and second data includingYoung's modulus (E), the yield strength (σy), and one of the stress atapproximately 29% plastic strain (σu) and the strain hardening exponent(η); and wherein said means for determining the stress in the stressedsection includesmeans for determining the hardness P_(ave) ; means fordetermining (h_(o))² =P_(s) {σy(1+σu/σy)C*(1+ln((tan α)E/3σy))}⁻¹ whereC* and tan α depend on the type of the indenter of the indentation test;means for determining the ratio R equivalent to R=(h_(s))² /(h_(o))² ;means for determining the stress in satisfaction of the followingformulae;

    if R<1; R={1+(geomf)stress/P.sub.ave }.sup.-1

    if R>1; R={1-stress/P.sub.ave }.sup.-1

where geomf=sin α, and α is related to the angle of indentation of theindenter of the indentation test.
 80. The apparatus of claim 69 whereinsaid means for determining stress includes means for determining stressfrom said first data including the indentation area A_(s) and seconddata including Young's modulus (E), the yield strength (σy), and one ofthe stress at approximately 29% plastic strain (σu) and the strainhardening exponent (η) and wherein said means for determining saidstress further includesmeans for determining the hardness P_(ave) meansfor determining (P_(o)) from P_(o) =(h_(s))2σy(1+σu/σy)C*{1+ln((tanα)E/3σy)}, where C* and tan α depend on the type of indenter used in theindentation test; means for determining the ratio R, where the means fordetermining R includes means for at least one of1) determining A_(o)from A_(o) =P_(o) /P_(ave) and determining R=A_(s) /A₀, where the firstdata includes A_(s) ; and 2) determining R=P_(s) /P₀, where the firstdata includes the load of indentation P_(s) means for determining thestress in satisfaction one of the following formulae

    if R<1; R={1+stress/P.sub.ave }.sup.-1

    if R>1; R={1-(geomf)stress/P.sub.ave }.sup.-1

where geomf=sin(α), and α is related to the angle of indentation of theindenter or the indentation test.
 81. An apparatus for determining thepreexisting stress in a stressed section of material, comprising:a dataprocessor, said data processor including program means for determiningthe stress in a stressed section of a material from first data and fromsecond data characteristic of the material, said first data obtainedfrom an indentation test on the stressed section and including at leasta loading (P) versus penetration (h) curve and said second dataincluding the yield strength of the material, said data processorfurther including means for fitting the loading to curve to a polynomialexpression of the form B₁ h² +B₂ h³ to determine at least first andsecond constants B₁ and B₂ means for determining at least one additionalconstant B₃ from B₁ ; and means for determining the stress G in thesection of material as a function of the at least one additionalconstant B₃ and the yield strength σy.
 82. The apparatus of claim 81wherein said means for determining at least one additional constantincludes means for determining B₃ according to the following formula B₃=B₁ /(K₁ (tan γ)²), where K₁ is a constant and γ is the angle ofindentation of the indenter used in the indentation test.
 83. Theapparatus of claim 81 wherein said means for determining the stress Gincludes means for determining the stress in accordance with thefollowing formula (G)² =(σy)² -(B₃)².
 84. The apparatus of claim 81including means for determining the change of stress with depth into thematerial.
 85. The apparatus of claim 84 wherein means for determiningthe change of stress with penetration includes means for determining afourth constant B₄ in satisfaction of the formula B₄ =B₂ /(K2(tan γ)³)and means for determining the magnitude of the slope of the stress insatisfaction of the formula G=-B₄ /G.
 86. The apparatus of claim 81wherein the means for determining at least one other constant includesmeans for determining B3 in satisfaction of the formula B₃ =B₁ /(K₁ (tanγ)²) and include means for determining a fourth additional constant B₄in satisfaction of the formula B₄ =B₂ /(K₂ (tan γ)³), where K₁ and K₂are constants and g depends on the indenter of the indentation test, andsaid means for determining the stress include means for determining thestress G in satisfaction of the formula (G)² =(sy)² -(B₃)² ; and whereinsaid apparatus further includesmeans for determining the sign of G; andmeans for determining Γ, the slope of change of the stress G withpenetration, in satisfaction of the formula Γ=-B₄ /G.
 87. The apparatusof claim 81 further including an indentation apparatus for indenting thestressed section of the material with an indenter to determine saidfirst data.
 88. Apparatus for determining the plastic strain in asection of material that has plastically yielded, comprising:a dataprocessor, said data processor including means for determining fromfirst data obtained by indenting the section with an indenter and fromsecond data characteristic of the material the stress in the stressedsection, the first data including a penetration (h) and a load (P) onthe indenter, said means including means for determining the residualyield strength as a function of the second data and the first data; andmeans for determining the plastic strain in the section of the materialfrom the residual yield strength and said second data.
 89. The apparatusof claim 88 wherein said means for determining the plastic strainincludes means for determining the plastic strain from second data thatincludes one of: the Young's modulus of the material and one of thestress at approximately 29% plastic strain and both of the strainhardening exponent and the yield strength.
 90. The apparatus of claim 88wherein said means for determining the plastic strain include means fordetermining the plastic strain from second data that includes astress-strain curve.
 91. The apparatus of claim 88 wherein said firstdata includes a depth of penetration hs at the load P_(s) ; andwhereinsaid second data includes the Young's modulus (E) and one of the strainhardening exponent (h) and the stress at approximately 29% plasticstrain of the material, and (σu); and wherein said means for determiningthe residual yield strength includes means for determining the residualyield strength from the Young's modulus, (E) the geometry of theindenter used in the indentation test, P_(s) and hs.
 92. The apparatusof claim 88 wherein determining the residual yield strength includesdetermining the residual yield strength based on first data including amaximum load Pm, a maximum depth of penetration hm and a residual depthof penetration hr from determined from unloading and based in seconddata including Young's modulus and one of the strain hardening exponentand the stress at approximately 29% plastic strain of the material. 93.The apparatus of claim 88 wherein said means for determining theresidual yield strength includes means for determining the residualyield strength from said first data including a slope upon initialunloading dP/dh and second data including Young's modulus (E) the areaAm corresponding to the maximum load on the indenter.
 94. The apparatusof claim 88 wherein said means for determining includes means fordetermining the Young's modulus of the material from first dataincluding an unloading slope and the area Am corresponding to themaximum load on the indenter, second data including the Poisson ratiothe material and the Young's modulus of the indenter.
 95. The apparatusof claim 88 wherein said means for de determining includes means fordetermining the Young's modulus of the material from first dataincluding an observed area of indentation, a maximum load Pm, a maximumpenetration hm and an unloading slope dP/dh and from second dataincluding the Poisson ratio of the material and from the Poisson ratioand the Young's modulus of the indenter of the indentation test, saidmeans for determining further including means for determining theresidual yield strength from the Young's modulus of the material, hr,hm, and Pm, an angle related to the angle of indentation of theindenter, and a constant determined by the indenter of the indentationtest.
 96. The aparatus of claim 88 wherein determining the residualplastic strain includes determining the residual yield strength σy insatisfaction of the following two equations:

    A.sub.m /(h.sub.m)2=9.96-12.64(1-Ho/E)+105.42(1-Ho/E).sup.2 =-229.57(1-Ho/E).sup.3 +157.67(1-Ho/E).sup.4              ( 1)

    σy=(σu)-0.29Ho                                 (2).


97. 97. The apparatus of claim 88 further including an indentationapparatus for indenting the stressed section of the material with anindenter to determine said first data.
 98. A method of determining thepreexisting stress in a stressed section of a material from dataobtainable via one indentation of the stressed section,comprising:obtaining first data from indentation of the stressedsection, the first data including at least two of the following: thearea of indentation, the depth of indentation, and the indentation load;and determining the preexisting stress as a function of the first dataand other data that is determinable other than via indentation with theindentor, and wherein the measurement of the deformation of the materialvia strain, tensiometric gauges or an extensometer placed in thevicinity of the area of contact between the indentor and the stressedsection is not required to determine the stress.
 99. A method ofdetermining the preexisting stress in a stressed section of a material,comprising:obtaining first data from the indentation of the stressedsection of the material with an indenter, the first data including atleast one of a) a depth of penetration and a load on the indentor and b)a depth of penetration and an area of indentation; obtaining second dataincluding at least one of 1) data obtained from an indentation of asecond section having a known stress of one of the material and a secondmaterial substantially similar to the material and 2) additional datathat can allow determination of the Young's modulus that ischaracteristic of the material, and determining from the first andsecond data the stress in the stressed section of material.
 100. Anapparatus for determining the preexisting stress in a stressed sectionof material, comprising:a data processor, said data processor includingprogram means for determining the stress in a stressed section of amaterial as a function of data including indentation data obtained fromfirst and second indentations to a substantially similar depth ofpenetration, one of the indentations being of the stressed section ofmaterial and the other of the indentations being of a second sectionhaving a known stress of one of the material and a substantially similarmaterial, the indentation data including at least one of a firstindentation load and a first indentation area of the first indentationand including at least one of a second indentation area and a secondindentation load of the second indentation, said data includingadditional data that can allow determination of the hardness of thematerial.
 101. An apparatus for determining the preexisting stress in astressed section of material, comprising:a data processor, said dataprocessor including program means for determining the stress in astressed section of a material as a function of data includingindentation data obtained from first and second to substantially thesame indentation load, the first indentation being of the stressedsection of the material and the second indentation being of a secondsection having a known stress of one of the material and a substantiallysimilar material, the indentation data including at least a first depthof indentation of the first indentation and a second depth ofindentation of the second indentation, said data including additionaldata including one of: ) the indentation load and the area of the secondindentation; and b) other data that can allow the determination of thehardness that is characteristic of the material.
 102. A method ofdetermining the preexisting stress in a stressed section of materialincluding using first and second indentations, comprising:obtainingindentation data from the first and second indentations to asubstantially similar depth of penetration, the first indentation beingof the stressed section of material and the second indentation being ofa second section having a known stress of one of the material and asubstantially similar material, the indentation data including at leastone of a first indentation load and a first indentation area of thefirst indentation and including at least one of a second indentationarea and a second indentation load of the second indentation; obtainingadditional data that can allow determination of the hardness of thematerial; and determining the stress from the data.
 103. The method ofclaim 102 wherein obtaining the indentation data includes obtaining thefirst indentation load and the second indentation load and whereinobtaining the additional data includes obtaining the second indentationarea.
 104. The method of claim 102 wherein obtaining the indentationdata includes obtaining the first indentation load and the secondindentation load and wherein obtaining the additional data includesobtaining the first indentation area.
 105. The method of claim 102wherein obtaining the indentation data includes obtaining the firstindentation load and the second indentation load.
 106. The method ofclaim 102 wherein obtaining the indentation data includes obtaining thefirst indentation area and the second indentation area and whereinobtaining the additional data includes obtaining the second indentationload.
 107. The method of claim 102 wherein obtaining the indentationdata includes obtaining the first indentation load and the firstindentation area and wherein obtaining the additional data includesobtaining the second indentation area.
 108. The method of claim 102wherein obtaining the indentation data includes obtaining the firstindentation area and the second indentation area.
 109. An apparatus fordetermining the preexisting stress in a stressed section of one of anelastoplastic material and a material exhibiting strain hardeningcomprising:a data processor, said data processor including program meansfor determining the stress in the stressed section of the one of anelastoplastic material and a material that can exhibit strain hardening,said program means including means for determining the stress at leastas a function of first and second areas of indentation obtained fromseparate indentations, one of the indentations being of the stressedsection of material and the other indentation being of a second section,having a known stress, of one of the material and a second material thatis substantially similar to the material, and as a function ofadditional data sufficient to allow the determination of the hardness ofthe material.